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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
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how do i calculate the probablity of getting a royal flush in the game of poker ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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how many triangles can be drawn by joining the vertices and centre of regular hexagon ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough .
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why is the part of the question : `` sorted however the layer chooses '' significant ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial .
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there 's only one 9 of spades.. so how can we choose 9 of spade first in the first selection and then in the secend selection 9 of spade last ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things .
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if we pick a spade , the probability of picking another spade increases , why do n't we have to worry about that affecting our result ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
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what is an undefined value ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting .
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surely if the order does n't matter then one can include all numbers with and without order ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial .
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it should be 9 cards numbered from 1-9 right ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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then , after the 1st card is chosen , for the second card are n't we have 36-4=32 choices left ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards .
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i had a question on my statistics homework that neither myself or my teacher could figure out the correct answer to : how many combinations are possible when choosing a hand of five cards from a standard deck of cards consisting of four cards from one suit and one card from another suit ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ?
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what is the probability of getting a multiple of 4 ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
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so what is the probability of a diamond flush ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards .
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what fraction of turns played on the machine will show five similar symbols in a row ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial .
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why do i have to divide by 9 ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them .
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by the way the question is asked would n't 4 be an acceptable answer meaning you can make 4 nine-card hands from 36 cards ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 .
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what would be the factorial of a negative number ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here .
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in terms of combinations vs permutations how do i decide whether a question takes into account order ( is it usually stated in the question ) ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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what are the possible permutations of a , a , b , b in four slots ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged .
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the number of ways of dividing 2p items into two equal groups of p each , groups having distinct identity ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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how many different hands can one draw in a game of 7-card rummy ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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how many unique combinations are possible for a word with 2 different letters repeating twice in the same word ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
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same question..if 3 cards were identical remaining 33 were unique..how does that change the answer ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting .
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so you use permutations when order matters and combinations when order does n't matter ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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just curious , does anyone know what the card game in the problem is ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 .
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without simplifying 36*35*34*33 ... ... ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount .
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what exactly is the factorial exclamation point thingy ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot .
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are there actually 94143280 different ways to split the 9 cards ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
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what 's the difference between permutation and combination ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting .
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i mean does the order in which the cards are grouped matter ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands .
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what is the probability that the sum is 30 ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order .
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is it possible to have n choose k if k is larger than n ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot .
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does n't it matter which way the cards were dealt , and how many people were playing ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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how many permutations can you make that are odd numbers ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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what do you do when you have a subtraction sign and addition sign ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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why does parentheses also work as a multiplication sign ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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when using the distributive property on a problem like 2 ( 4+b ) , would your answer be 8+2b or would it be 6b ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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is it possible to have the distributive law of division over addition/subtraction/multiplication ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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why when using the distributive law we have to multiply vs any of the other concepts ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ?
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would n't ( 45-20 ) be the real answer insted of 25 ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 .
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like is n't distrubitve property meaning only mutiple the numbers in the parenthiess only ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property .
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what can be the way of finding the value of the variable is unknown ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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what is the difference between pemdas and bodmas ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 .
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what happens if i do n't know a number ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 .
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what would happen if the distributed number was a letter ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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what was the simplified expression ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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is ( a multiplication sign ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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what is the point of using distributive property if you can just use pemdas ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 .
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how would you use two digit numbers ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
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why did sal use the dot does that mean mulitpycation ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
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how do you use the distributive property when dealing with fractions ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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why is the mutiplucation symble a dot not a `` x '' ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color .
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when sal does 5 ( 9-4 ) why is n't a multiplication sighn next to the perenthasies ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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what is the reason of the distributive law ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
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it what scenario would i ever want to use the distributive law of multication ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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how many multiplication signs are there ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it .
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what does the question mean by simplify ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
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what if 3 variables are unknown ?
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rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 . in the first distributive property video , we gave you an idea of why you have to distribute the 5 , why it makes sense , why you do n't just multiply it by the 9 . and we 're going to verify that it gives us the same answer as if we just evaluated the 9 minus 4 first . but anyway , what are these things ? so 5 times 9 , that is 45 . so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color . 5 times 9 minus 4 . so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
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you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
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also , what does `` distribute '' literally mean ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences .
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can a word be a phobia ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia .
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what 's the fear of parasites called ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia .
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is acrophobia ( fear of heights ) common ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety .
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what is the phobia of being afraid of singing ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo .
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anxiety disorder is same as phobia ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 .
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how do people get a fear of moths ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia .
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what is the name for the fear of sharp things ?
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phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried about flying on a plane . people with specific phobias work super hard to avoid whatever object they 're afraid of even is they know there 's actually no real danger or threat . and they might feel powerless to stop this fear and feel extreme anxiety even thinking about the object . because of this , phobias can seriously disrupt daily routines , limit your work efficiency , reduce your self-esteem , and strain relationships because someone might do whatever they can to avoid feeling the anxious and sometimes terrifying feelings of a phobia . as an example , many people might feel little uneasier or annoyed camping at night when tons of moths start to swarm around your lanterns . but , knowing this will probably happen likely wo n't affect your decision to go camping , right ? someone with mottephobia or fear of moths might actually avoid the camping trip altogether because they know that there 's this chance of seeing a moth . this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spiders or arachnophobia . an example situation could be like claustrophobia , which is a fear of constricted or enclosed , or tight spaces . someone with claustrophobia might have this intense fear of getting on an elevator because it 's so compact . as somewhat of an opposite example , agoraphobia is a fear of going out into an open or crowded space which can be anything from malls to markets , to theaters . oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia . now being exposed to the feared object or even thinking about being exposed to the feared object might cause severe symptoms of anxiety that are much stronger than the real threat . people with phobias often catastrophize , or immediately jump to the worst case scenario and also overestimate the chances of that actually happening . this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an everyday anxiety might be feeling queezy when you enter the spider exhibit at the zoo . someone with arachnophobia or fear of spiders might avoid going to the zoo altogether to avoid seeing any spiders . another everyday anxiety might be feeling dizzy when climbing a ladder . someone with a fear of heights might not go to their best friend 's wedding just because it 's on the 30th floor of a hotel . finally , an everyday anxiety might be being scared on a plane during severe turbulence where someone with aerophobia might avoid getting on a plane altogether , even if it was to go accept a promotion to their job . if someone does actually have a phobia and it 's not just everyday fear , what causes it ? where does this come from ? well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences . for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your symptoms as well having your physical exam taken and asking about particular medications which can all help rule out other medical conditions that might be producing the anxiety . once the phobia is diagnosed , though , treatment can be administered . here it 's important to tailor the treatment to the specific patients since patients respond differently to therapy , especially if other conditions are involved like depression or drug abuse . cognitive behavior therapy is specific kind of psychotherapy is particularly effective for specific phobias . first , therapists will likely try to help the patient identify a mistake in beliefs and realize they 're probably overestimating their fear and talk about the realistic risks . for example , being bitten by a rabid dog is actually a pretty rare occurrence . secondly , they 'll expose the person to the feared situation . using the same example , the therapist might have the patient get closer to a dog which helps them learn to take acceptable risks that are relatively safe . although the treatment regimen varies with each patient and the time required for treatment also varies , the vast majority of people with specific phobias can be helped with professional care .
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oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia .
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if someone had a family member with a fear , would another family member develop a fear of the same thing , or is it possible to develop a fear of something else because a family member had a fear ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with .
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why does eve make an effort to 'cover ' herself , but adam only covers his face ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real .
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what is `` modeling '' ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion .
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since many people were illiterate during this period , would these paintings have been considered a tool to help people grasp what they have leaned orally ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century .
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why is there a different shade around adam and eve ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup .
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did the artist intend that ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work .
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does oxidized silver remain silver or does oxidation convert it to a different element ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry .
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why would n't masaccio do a better job of hiding the seams of the giornata around the contours of the figures instead of so obvious a halo surrounding them ?
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( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion . we could ask what is the expulsion doing here ? this is the story of adam and eve being expelled from the garden of eden . they 've eaten the forbidden fruit from the tree of knowledge and god has discovered that transgression and has banished them from eden and we see a foreshortened angel . dr. zucker : that 's an armed angel , it looks like the marshall to me . dr. harris : chasing them out of the garden of eden . dr. zucker : their being evicted . dr. harris : what follows from this is that mankind knows then and ... dr. zucker : and death . dr. harris : exactly . this is the moment from which everything else comes in terms of catholic understanding of man 's destiny . dr. zucker : that 's right because it is from this fall from grace that christ is required . dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that masaccio has to people like giotto in the previous century where architecture is sometimes used , simply as a foil , as a kind of stage set . dr. harris : there 's so much emotion . dr. zucker : i 'm especially interested in the contrast of emotion . adam is covering his face , there is a kind of shame and a real awareness of his sin . his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it . dr. zucker : that 's right , long after . but we 've been restored to the original nudity that masaccio gave us , which is absolutely era appropriate , but he 's not covering his body , he 's covering his face ; it 's a kind of internal sense of guilt . whereas eve seems to have been taken directly from the ancient classical prototype of the modest venus . she 's shown in a beautiful contrapposto covering herself , but it 's her shame which seems more physical , but because her face is exposed we can see the real pain that she expresses through it . dr. harris : you said beautiful contrapposto , but i think about contrapposto as a standing , relaxed pose and these figures are in motion . dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup . dr. harris : and as a result he has n't quite gotten all of it perfectly . dr. zucker : no , there 's some awkward passages there . dr. harris : yeah , adam 's arms are a little bit too short , eve 's left arm is a little bit too long . given that masaccio 's the first artist to really attempt this naturalism in 1,000 years , some of that is to be forgiven . dr. zucker : i have to say that i think he 's done an extraordinary job . if you look at adam 's abdomen , for example , it is really beautifully rendered . there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and also that foreshortened angel is helping to create a sense of space for the figures to exist in , even though , as you pointed out , that architecture is more symbolic than real . dr. zucker : yeah , it 's just totally schematic is n't it ? dr. harris : yeah . dr. zucker : a couple of changes that are probably worth noting . one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century . dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plaster would otherwise dry . dr. zucker : other changes that have taken place in the painting that i think are worth noting are that the sword and the rays of light that are emanating from eden are now black , but that 's oxidized silver and it would have been very shiny initially . i think it 's importantalso to note that the expulsion is the first scene that we look at as we enter into this chapel , they literally walk into this story . almost like a panel in the cartoon it is leading our eye from left to right so that we can read through this story of st. peter . ( piano playing )
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his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it .
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why was the nudity covered up ?
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( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impressionist painting . male : in fact , the painting that we 're looking at is a painting that is about what we are doing right now , that is , looking at paintings . female : ( chuckles ) right . i almost feel as though i 'm looking in a mirror when i look up at this painting . male : in fact , we 're in the museum of fine arts in boston , looking at edgar degas ' visit to a museum . female : we think that the painting shows mary cassatt , degas ' friend because degas actually did a series of prints of mary cassatt in two different galleries in the louvre . he also did another oil painting of this subject . he also did pastels , so it was clearly one that he really liked , but he did that with his subjects . once he found a subject , like the races , or the ballet , or the nude , he would visit it time and time again and rework it . he clearly did that with this subject . male : in fact , art historians think that perhaps the woman who 's seated is mary cassatt 's sister , lydia . female : degas and mary cassatt were friends for 40 years and degas actually was the one who suggested that mary cassatt exhibit with the impressionist group , which she did . she was the only american to do so . male : they were really like-minded . it 's interesting because degas is often criticized as being a misogynist , somebody who has a very difficult time with women ; but it 's a more complex issue and it has to do also with class . mary cassatt had come from a very wealthy american family , and so in some ways , although she did not have the pedigree that degas ' family name had , they were on par . female : the story is that mary cassatt saw a degas in a shop window and pressed her nose against the window so that she could see it and take it in and really admired it ; and that similarly degas saw a painting by mary cassatt in 1874 and really admired it and saw her as a kindred spirit . male : in fact , degas is quoted saying to mary cassatt , `` most women paint as though they are `` trimming hats . not you . '' so they had this really close relationship . cassatt is reported to have stepped in when models were not able to do exactly what degas wanted them to do . female : degas took mary cassatt as his model for this subject of at the museum because by doing so he was immediately taking a woman who was interested in looking at art and who had a really good eye , and a really intelligent eye . male : look at the pose that he 's placed her in . her head is cocked back . she is really assessing and there is a sense of the discerning here . female : it 's interesting , this balance of looking at printed material and looking at visual material that 's on the walls of the gallery . that 's something that we see around us if we just look around this gallery that we 're in today . male : this is a painting that 's in some ways about the way we read the visual . it 's a painting about seeing . the subject is somebody looking at art as we , the viewers in the museum , are looking at art , so there 's this perfect replication . female : but more than looking , mary cassett is standing , tilting , gesturing , moving . so is her companion , holding the book , looking up , tilting her head . i think that 's precisely what degas was interested in , the same way that he 's interesting in gesture if we look at other subjects that he found compelling , like milliners , or ballet dancers ; the way that they stand , the way that they move . i think there was something really interesting and modern to degas about the way people move , and look at art , and gesture , and stand in an art museum . male : right . it 's an expression of the modern world . it 's interesting and , i think , instructive to think about what degas has not focused on . he has not shown us what the specific works of art are that she is looking at . we ca n't even see the body or the dress all that clearly . the entire painting is beautifully brushy and it gives a sense of the momentary . female : that 's right . if anything is specific it 's the gesture , it 's the posture . i think that degas is very much thinking about the kinds of questions that baudelaire raised , putting away the rhetorical gestures of history painting , of images of ancient greece and rome that we think of when we think about salon paintings by couture or gerome or bouguereau . for degas ' viewers , this must have looked ugly . these are n't the graceful gestures of painting condoned by the royal academy . these are really modern and , in some ways , ugly gestures ; the way you might capture someone sitting on the subway . male : all the paint is really quite muddy as well . the colors are muddy . the composition is also at odds with what academic painting would expect . they 're looking to the upper right , but the painting 's construction , the orthogonals , look , for instance , at the bench , the way the plane of the floor meets the plane of the wall , all of that rushes to the upper left , so they 're in opposition to each other . there 's this real sense of movement and energy in the painting that 's created by this compositional opposition that he 's constructed , but it is so much at odds with the way that an academic painting would be organized . female : right . in an academic painting we would likely see what they were looking at and we would see their expressions and there would be some narrative to be constructed out of that , but that 's not what interested degas . ( jazzy music )
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male : in fact , art historians think that perhaps the woman who 's seated is mary cassatt 's sister , lydia . female : degas and mary cassatt were friends for 40 years and degas actually was the one who suggested that mary cassatt exhibit with the impressionist group , which she did . she was the only american to do so .
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why is it that monet , picasso , degas , and van gogh are artists that everybody knows something about but artists like mary cassat , winslow homer , casper david friedrich , and rosa bonheur are just as good but less well known ( i for one had n't heard of any of them til i watched these ) ?
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( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impressionist painting . male : in fact , the painting that we 're looking at is a painting that is about what we are doing right now , that is , looking at paintings . female : ( chuckles ) right . i almost feel as though i 'm looking in a mirror when i look up at this painting . male : in fact , we 're in the museum of fine arts in boston , looking at edgar degas ' visit to a museum . female : we think that the painting shows mary cassatt , degas ' friend because degas actually did a series of prints of mary cassatt in two different galleries in the louvre . he also did another oil painting of this subject . he also did pastels , so it was clearly one that he really liked , but he did that with his subjects . once he found a subject , like the races , or the ballet , or the nude , he would visit it time and time again and rework it . he clearly did that with this subject . male : in fact , art historians think that perhaps the woman who 's seated is mary cassatt 's sister , lydia . female : degas and mary cassatt were friends for 40 years and degas actually was the one who suggested that mary cassatt exhibit with the impressionist group , which she did . she was the only american to do so . male : they were really like-minded . it 's interesting because degas is often criticized as being a misogynist , somebody who has a very difficult time with women ; but it 's a more complex issue and it has to do also with class . mary cassatt had come from a very wealthy american family , and so in some ways , although she did not have the pedigree that degas ' family name had , they were on par . female : the story is that mary cassatt saw a degas in a shop window and pressed her nose against the window so that she could see it and take it in and really admired it ; and that similarly degas saw a painting by mary cassatt in 1874 and really admired it and saw her as a kindred spirit . male : in fact , degas is quoted saying to mary cassatt , `` most women paint as though they are `` trimming hats . not you . '' so they had this really close relationship . cassatt is reported to have stepped in when models were not able to do exactly what degas wanted them to do . female : degas took mary cassatt as his model for this subject of at the museum because by doing so he was immediately taking a woman who was interested in looking at art and who had a really good eye , and a really intelligent eye . male : look at the pose that he 's placed her in . her head is cocked back . she is really assessing and there is a sense of the discerning here . female : it 's interesting , this balance of looking at printed material and looking at visual material that 's on the walls of the gallery . that 's something that we see around us if we just look around this gallery that we 're in today . male : this is a painting that 's in some ways about the way we read the visual . it 's a painting about seeing . the subject is somebody looking at art as we , the viewers in the museum , are looking at art , so there 's this perfect replication . female : but more than looking , mary cassett is standing , tilting , gesturing , moving . so is her companion , holding the book , looking up , tilting her head . i think that 's precisely what degas was interested in , the same way that he 's interesting in gesture if we look at other subjects that he found compelling , like milliners , or ballet dancers ; the way that they stand , the way that they move . i think there was something really interesting and modern to degas about the way people move , and look at art , and gesture , and stand in an art museum . male : right . it 's an expression of the modern world . it 's interesting and , i think , instructive to think about what degas has not focused on . he has not shown us what the specific works of art are that she is looking at . we ca n't even see the body or the dress all that clearly . the entire painting is beautifully brushy and it gives a sense of the momentary . female : that 's right . if anything is specific it 's the gesture , it 's the posture . i think that degas is very much thinking about the kinds of questions that baudelaire raised , putting away the rhetorical gestures of history painting , of images of ancient greece and rome that we think of when we think about salon paintings by couture or gerome or bouguereau . for degas ' viewers , this must have looked ugly . these are n't the graceful gestures of painting condoned by the royal academy . these are really modern and , in some ways , ugly gestures ; the way you might capture someone sitting on the subway . male : all the paint is really quite muddy as well . the colors are muddy . the composition is also at odds with what academic painting would expect . they 're looking to the upper right , but the painting 's construction , the orthogonals , look , for instance , at the bench , the way the plane of the floor meets the plane of the wall , all of that rushes to the upper left , so they 're in opposition to each other . there 's this real sense of movement and energy in the painting that 's created by this compositional opposition that he 's constructed , but it is so much at odds with the way that an academic painting would be organized . female : right . in an academic painting we would likely see what they were looking at and we would see their expressions and there would be some narrative to be constructed out of that , but that 's not what interested degas . ( jazzy music )
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( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day .
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what causes an artist to become famous ?
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( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impressionist painting . male : in fact , the painting that we 're looking at is a painting that is about what we are doing right now , that is , looking at paintings . female : ( chuckles ) right . i almost feel as though i 'm looking in a mirror when i look up at this painting . male : in fact , we 're in the museum of fine arts in boston , looking at edgar degas ' visit to a museum . female : we think that the painting shows mary cassatt , degas ' friend because degas actually did a series of prints of mary cassatt in two different galleries in the louvre . he also did another oil painting of this subject . he also did pastels , so it was clearly one that he really liked , but he did that with his subjects . once he found a subject , like the races , or the ballet , or the nude , he would visit it time and time again and rework it . he clearly did that with this subject . male : in fact , art historians think that perhaps the woman who 's seated is mary cassatt 's sister , lydia . female : degas and mary cassatt were friends for 40 years and degas actually was the one who suggested that mary cassatt exhibit with the impressionist group , which she did . she was the only american to do so . male : they were really like-minded . it 's interesting because degas is often criticized as being a misogynist , somebody who has a very difficult time with women ; but it 's a more complex issue and it has to do also with class . mary cassatt had come from a very wealthy american family , and so in some ways , although she did not have the pedigree that degas ' family name had , they were on par . female : the story is that mary cassatt saw a degas in a shop window and pressed her nose against the window so that she could see it and take it in and really admired it ; and that similarly degas saw a painting by mary cassatt in 1874 and really admired it and saw her as a kindred spirit . male : in fact , degas is quoted saying to mary cassatt , `` most women paint as though they are `` trimming hats . not you . '' so they had this really close relationship . cassatt is reported to have stepped in when models were not able to do exactly what degas wanted them to do . female : degas took mary cassatt as his model for this subject of at the museum because by doing so he was immediately taking a woman who was interested in looking at art and who had a really good eye , and a really intelligent eye . male : look at the pose that he 's placed her in . her head is cocked back . she is really assessing and there is a sense of the discerning here . female : it 's interesting , this balance of looking at printed material and looking at visual material that 's on the walls of the gallery . that 's something that we see around us if we just look around this gallery that we 're in today . male : this is a painting that 's in some ways about the way we read the visual . it 's a painting about seeing . the subject is somebody looking at art as we , the viewers in the museum , are looking at art , so there 's this perfect replication . female : but more than looking , mary cassett is standing , tilting , gesturing , moving . so is her companion , holding the book , looking up , tilting her head . i think that 's precisely what degas was interested in , the same way that he 's interesting in gesture if we look at other subjects that he found compelling , like milliners , or ballet dancers ; the way that they stand , the way that they move . i think there was something really interesting and modern to degas about the way people move , and look at art , and gesture , and stand in an art museum . male : right . it 's an expression of the modern world . it 's interesting and , i think , instructive to think about what degas has not focused on . he has not shown us what the specific works of art are that she is looking at . we ca n't even see the body or the dress all that clearly . the entire painting is beautifully brushy and it gives a sense of the momentary . female : that 's right . if anything is specific it 's the gesture , it 's the posture . i think that degas is very much thinking about the kinds of questions that baudelaire raised , putting away the rhetorical gestures of history painting , of images of ancient greece and rome that we think of when we think about salon paintings by couture or gerome or bouguereau . for degas ' viewers , this must have looked ugly . these are n't the graceful gestures of painting condoned by the royal academy . these are really modern and , in some ways , ugly gestures ; the way you might capture someone sitting on the subway . male : all the paint is really quite muddy as well . the colors are muddy . the composition is also at odds with what academic painting would expect . they 're looking to the upper right , but the painting 's construction , the orthogonals , look , for instance , at the bench , the way the plane of the floor meets the plane of the wall , all of that rushes to the upper left , so they 're in opposition to each other . there 's this real sense of movement and energy in the painting that 's created by this compositional opposition that he 's constructed , but it is so much at odds with the way that an academic painting would be organized . female : right . in an academic painting we would likely see what they were looking at and we would see their expressions and there would be some narrative to be constructed out of that , but that 's not what interested degas . ( jazzy music )
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male : this is a painting that 's in some ways about the way we read the visual . it 's a painting about seeing . the subject is somebody looking at art as we , the viewers in the museum , are looking at art , so there 's this perfect replication .
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what is the name of that painting ?
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( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impressionist painting . male : in fact , the painting that we 're looking at is a painting that is about what we are doing right now , that is , looking at paintings . female : ( chuckles ) right . i almost feel as though i 'm looking in a mirror when i look up at this painting . male : in fact , we 're in the museum of fine arts in boston , looking at edgar degas ' visit to a museum . female : we think that the painting shows mary cassatt , degas ' friend because degas actually did a series of prints of mary cassatt in two different galleries in the louvre . he also did another oil painting of this subject . he also did pastels , so it was clearly one that he really liked , but he did that with his subjects . once he found a subject , like the races , or the ballet , or the nude , he would visit it time and time again and rework it . he clearly did that with this subject . male : in fact , art historians think that perhaps the woman who 's seated is mary cassatt 's sister , lydia . female : degas and mary cassatt were friends for 40 years and degas actually was the one who suggested that mary cassatt exhibit with the impressionist group , which she did . she was the only american to do so . male : they were really like-minded . it 's interesting because degas is often criticized as being a misogynist , somebody who has a very difficult time with women ; but it 's a more complex issue and it has to do also with class . mary cassatt had come from a very wealthy american family , and so in some ways , although she did not have the pedigree that degas ' family name had , they were on par . female : the story is that mary cassatt saw a degas in a shop window and pressed her nose against the window so that she could see it and take it in and really admired it ; and that similarly degas saw a painting by mary cassatt in 1874 and really admired it and saw her as a kindred spirit . male : in fact , degas is quoted saying to mary cassatt , `` most women paint as though they are `` trimming hats . not you . '' so they had this really close relationship . cassatt is reported to have stepped in when models were not able to do exactly what degas wanted them to do . female : degas took mary cassatt as his model for this subject of at the museum because by doing so he was immediately taking a woman who was interested in looking at art and who had a really good eye , and a really intelligent eye . male : look at the pose that he 's placed her in . her head is cocked back . she is really assessing and there is a sense of the discerning here . female : it 's interesting , this balance of looking at printed material and looking at visual material that 's on the walls of the gallery . that 's something that we see around us if we just look around this gallery that we 're in today . male : this is a painting that 's in some ways about the way we read the visual . it 's a painting about seeing . the subject is somebody looking at art as we , the viewers in the museum , are looking at art , so there 's this perfect replication . female : but more than looking , mary cassett is standing , tilting , gesturing , moving . so is her companion , holding the book , looking up , tilting her head . i think that 's precisely what degas was interested in , the same way that he 's interesting in gesture if we look at other subjects that he found compelling , like milliners , or ballet dancers ; the way that they stand , the way that they move . i think there was something really interesting and modern to degas about the way people move , and look at art , and gesture , and stand in an art museum . male : right . it 's an expression of the modern world . it 's interesting and , i think , instructive to think about what degas has not focused on . he has not shown us what the specific works of art are that she is looking at . we ca n't even see the body or the dress all that clearly . the entire painting is beautifully brushy and it gives a sense of the momentary . female : that 's right . if anything is specific it 's the gesture , it 's the posture . i think that degas is very much thinking about the kinds of questions that baudelaire raised , putting away the rhetorical gestures of history painting , of images of ancient greece and rome that we think of when we think about salon paintings by couture or gerome or bouguereau . for degas ' viewers , this must have looked ugly . these are n't the graceful gestures of painting condoned by the royal academy . these are really modern and , in some ways , ugly gestures ; the way you might capture someone sitting on the subway . male : all the paint is really quite muddy as well . the colors are muddy . the composition is also at odds with what academic painting would expect . they 're looking to the upper right , but the painting 's construction , the orthogonals , look , for instance , at the bench , the way the plane of the floor meets the plane of the wall , all of that rushes to the upper left , so they 're in opposition to each other . there 's this real sense of movement and energy in the painting that 's created by this compositional opposition that he 's constructed , but it is so much at odds with the way that an academic painting would be organized . female : right . in an academic painting we would likely see what they were looking at and we would see their expressions and there would be some narrative to be constructed out of that , but that 's not what interested degas . ( jazzy music )
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( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day .
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i may be jumping the gun here , but has the term `` impressionism '' come about by the time this painting was painted ?
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alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt at an invasion , and once again , was turned away by the forces of union general , george meade . gettysburg was the most destructive battle of the civil war , there were about 50,000 casualties , and it , along with the victory of the siege of vicksburg , which followed the day after , on july 4th , really start to signify the beginning of the end of the confederacy 's bid for independence . now , what you may not know about the battle of gettysburg , is that is was almost the end of the war , in fact , lee took his army , trying to cross back over the potomac into the south , and the potomac was flooded so , he and his army were pretty much pinned , between this flooded river , and the forces of meade in the north . now meade , if he had attacked , probably could 've won the war right there and then , and lincoln was so angry that meade did n't attack , he wrote him this really nasty letter saying , `` i think you do n't even realize what you 've done here by letting lee get away , we could 've ended the war right now '' , but actually lincoln did n't send that letter , he thought better of it , and instead congratulated meade on his great victory and , the boost of morale that it gave the forces of the united states , at gettysburg . so now i 'd like to take some time to talk about the gettysburg address , which is arguably , the most famous speech in american history , it 's pretty up there . and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg . i think the gettysburg address is really interesting , and all of the events surrounding it , the circumstances surrounding it , tell us a lot about the culture and society of the 19th century , the progress of the civil war , and also the way that things are going to kind of be wrapped up in the end of the civil war . what the ultimate message of the war is going to be , and what the blueprint of reuniting the country is going to look like . so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the governor of pennsylvania , contracts out to create a cemetery . and in this period of three and a half months , there are bodies literally rotting on the ground , so it 's a bit of a hellscape , the entire town of gettysburg stinks , they had to burn all the dead horses , so it smells like burning horses , and rotting human flesh . it is not a happy place to be . so the town of gettysburg and the state of pennsylvania , are very eager to get a cemetery underway at gettysburg . and so they begin the process of burying the bodies , and re-burying the bodies , trying to identify the various corpses that are left on the fields . and they ask this man , edward everett , who is really the preeminent orator of his day . he was like the rock concert of the 19th century , to come and give an oration on the dedication of the gettysburg cemetery . and they say , `` everett , do you think you can do this on october 23rd ? '' and everett says , `` no , i definitely wo n't be ready to have a script for an oration by then , so can you push it back to november 19th ? '' so , it 's actually everett who decides , what day the gettysburg address is going to take place on . lincoln , by contrast , was only invited maybe a month or so before , and he was n't really considered the important speaker of the day , that was everett . but lincoln knew , that he wanted to make something of his remarks at gettysburg . now remember that , an election year is coming up in 1864 , it 's been a hard year , gettysburg is the first major victory that the united states forces have had in a long time , so he kinda wants to make sure that he can set the tone of how gettysburg is going to be remembered , and to reconfirm a sense of mission , about the civil war right , when there 's been such a great loss of life , and when you 're standing around looking at that loss of life , it can be very easy to get discouraged and say , okay , maybe we should just end the war , we should have peace now , allow the south to secede and , retain slavery , and lincoln wants to make sure that people come away from this dedication at gettysburg , with a renewed sense of purpose , in continuing to fight the civil war . now there 's a common misconception that , lincoln wrote the gettysburg address on the back of an envelope on the train to gettysburg . that is almost certainly not the case , because lincoln was a planner . remember that he was self-educated , and he always took a lot of time in anything that he wrote . he wrote drafts and got revisions and , wrote yet another draft . he liked to be extremely precise with his language , and you can see that throughout pretty much everything that he 's written , that he is an extremely effective and eloquent writer . and that was n't just because he was an extremely eloquent person , he was . that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day . now edward everett was i think , the undisputed champion of giving speeches in his day . he was an incredible speaker , and everyone who was there , actually agreed that , everett did an incredible job speaking , he spoke for over two hours , and if that sounds like a really long time to us , for the 19th century that was actually pretty appropriate . that 's what people expected out of oratory , in the 19th century . they paid attention , they were riveted by it . it was like going to see a movie or a concert today . so people really wanted to hear everett talk for that long . in fact , they were quite confused , when lincoln did n't talk for longer then just a couple of minutes . a lot of people even were reported to say , `` was that it ? '' so here in the center we have a picture of the day at gettysburg and , we 're pretty sure that this is the only confirmed picture of lincoln at gettysburg . now , he 's kinda small here , but i think this is a really interesting picture , cuz it gives you a sense of what lincoln 's stature was at the time , and also the people that he surrounded himself with . so , this is lincoln here , right here in the center , not wearing a hat , looking down , and then he 's surrounded by the important people of his cabinet , so right here , i 'm pretty sure this is william seward , who was the secretary of state , and over here , these are john hay and john nicolay who were lincoln 's personal secretaries . they went everywhere with him , and this guy up here is a little harder to see , that is edward everett . now imagine what it would have been like , to stand on this field , in this growing cemetery at gettysburg , and listen to edward everett , and abraham lincoln talk about the meaning of the battle around you . now remember , that it 's november , so it 's been three and a half months since the battle , but the battle of gettysburg took place , in the beginning of july , and it was 90 , 100 degrees outside , so when lee and meade left gettysburg , they left 8,000 or more bodies , rotting in the hot july sun . and many of them had been out there rotting , for those three months so , when you were standing on this field at gettysburg , there would have literally been human bones around you , that you could see . it probably would have still smelled pretty terrible , so you 're really kind of in the thick of the destruction of the civil war , and listening to these two men , who are trying to make meaning out of it for you , so everett gets up , and he gives this fiery speech for two hours , and he goes through all of the details of the battle and says , `` this is what happened over on that hill , and this is what happened over on that hill '' , and he tries to rev up the crowd into kind of this patriotic fervor , of not only appreciating the glory of the union victory at gettysburg , but also renewing their hatred for their enemy , and then lincoln gets up to speak , and he speaks for just a couple minutes , and we 'll talk more about that , in the next video .
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and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties .
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why did meade remain so passive at such a crucial moment ?
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alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt at an invasion , and once again , was turned away by the forces of union general , george meade . gettysburg was the most destructive battle of the civil war , there were about 50,000 casualties , and it , along with the victory of the siege of vicksburg , which followed the day after , on july 4th , really start to signify the beginning of the end of the confederacy 's bid for independence . now , what you may not know about the battle of gettysburg , is that is was almost the end of the war , in fact , lee took his army , trying to cross back over the potomac into the south , and the potomac was flooded so , he and his army were pretty much pinned , between this flooded river , and the forces of meade in the north . now meade , if he had attacked , probably could 've won the war right there and then , and lincoln was so angry that meade did n't attack , he wrote him this really nasty letter saying , `` i think you do n't even realize what you 've done here by letting lee get away , we could 've ended the war right now '' , but actually lincoln did n't send that letter , he thought better of it , and instead congratulated meade on his great victory and , the boost of morale that it gave the forces of the united states , at gettysburg . so now i 'd like to take some time to talk about the gettysburg address , which is arguably , the most famous speech in american history , it 's pretty up there . and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg . i think the gettysburg address is really interesting , and all of the events surrounding it , the circumstances surrounding it , tell us a lot about the culture and society of the 19th century , the progress of the civil war , and also the way that things are going to kind of be wrapped up in the end of the civil war . what the ultimate message of the war is going to be , and what the blueprint of reuniting the country is going to look like . so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the governor of pennsylvania , contracts out to create a cemetery . and in this period of three and a half months , there are bodies literally rotting on the ground , so it 's a bit of a hellscape , the entire town of gettysburg stinks , they had to burn all the dead horses , so it smells like burning horses , and rotting human flesh . it is not a happy place to be . so the town of gettysburg and the state of pennsylvania , are very eager to get a cemetery underway at gettysburg . and so they begin the process of burying the bodies , and re-burying the bodies , trying to identify the various corpses that are left on the fields . and they ask this man , edward everett , who is really the preeminent orator of his day . he was like the rock concert of the 19th century , to come and give an oration on the dedication of the gettysburg cemetery . and they say , `` everett , do you think you can do this on october 23rd ? '' and everett says , `` no , i definitely wo n't be ready to have a script for an oration by then , so can you push it back to november 19th ? '' so , it 's actually everett who decides , what day the gettysburg address is going to take place on . lincoln , by contrast , was only invited maybe a month or so before , and he was n't really considered the important speaker of the day , that was everett . but lincoln knew , that he wanted to make something of his remarks at gettysburg . now remember that , an election year is coming up in 1864 , it 's been a hard year , gettysburg is the first major victory that the united states forces have had in a long time , so he kinda wants to make sure that he can set the tone of how gettysburg is going to be remembered , and to reconfirm a sense of mission , about the civil war right , when there 's been such a great loss of life , and when you 're standing around looking at that loss of life , it can be very easy to get discouraged and say , okay , maybe we should just end the war , we should have peace now , allow the south to secede and , retain slavery , and lincoln wants to make sure that people come away from this dedication at gettysburg , with a renewed sense of purpose , in continuing to fight the civil war . now there 's a common misconception that , lincoln wrote the gettysburg address on the back of an envelope on the train to gettysburg . that is almost certainly not the case , because lincoln was a planner . remember that he was self-educated , and he always took a lot of time in anything that he wrote . he wrote drafts and got revisions and , wrote yet another draft . he liked to be extremely precise with his language , and you can see that throughout pretty much everything that he 's written , that he is an extremely effective and eloquent writer . and that was n't just because he was an extremely eloquent person , he was . that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day . now edward everett was i think , the undisputed champion of giving speeches in his day . he was an incredible speaker , and everyone who was there , actually agreed that , everett did an incredible job speaking , he spoke for over two hours , and if that sounds like a really long time to us , for the 19th century that was actually pretty appropriate . that 's what people expected out of oratory , in the 19th century . they paid attention , they were riveted by it . it was like going to see a movie or a concert today . so people really wanted to hear everett talk for that long . in fact , they were quite confused , when lincoln did n't talk for longer then just a couple of minutes . a lot of people even were reported to say , `` was that it ? '' so here in the center we have a picture of the day at gettysburg and , we 're pretty sure that this is the only confirmed picture of lincoln at gettysburg . now , he 's kinda small here , but i think this is a really interesting picture , cuz it gives you a sense of what lincoln 's stature was at the time , and also the people that he surrounded himself with . so , this is lincoln here , right here in the center , not wearing a hat , looking down , and then he 's surrounded by the important people of his cabinet , so right here , i 'm pretty sure this is william seward , who was the secretary of state , and over here , these are john hay and john nicolay who were lincoln 's personal secretaries . they went everywhere with him , and this guy up here is a little harder to see , that is edward everett . now imagine what it would have been like , to stand on this field , in this growing cemetery at gettysburg , and listen to edward everett , and abraham lincoln talk about the meaning of the battle around you . now remember , that it 's november , so it 's been three and a half months since the battle , but the battle of gettysburg took place , in the beginning of july , and it was 90 , 100 degrees outside , so when lee and meade left gettysburg , they left 8,000 or more bodies , rotting in the hot july sun . and many of them had been out there rotting , for those three months so , when you were standing on this field at gettysburg , there would have literally been human bones around you , that you could see . it probably would have still smelled pretty terrible , so you 're really kind of in the thick of the destruction of the civil war , and listening to these two men , who are trying to make meaning out of it for you , so everett gets up , and he gives this fiery speech for two hours , and he goes through all of the details of the battle and says , `` this is what happened over on that hill , and this is what happened over on that hill '' , and he tries to rev up the crowd into kind of this patriotic fervor , of not only appreciating the glory of the union victory at gettysburg , but also renewing their hatred for their enemy , and then lincoln gets up to speak , and he speaks for just a couple minutes , and we 'll talk more about that , in the next video .
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alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt at an invasion , and once again , was turned away by the forces of union general , george meade .
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do we have a `` fog of war '' moment with inadequate intelligence ?
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alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt at an invasion , and once again , was turned away by the forces of union general , george meade . gettysburg was the most destructive battle of the civil war , there were about 50,000 casualties , and it , along with the victory of the siege of vicksburg , which followed the day after , on july 4th , really start to signify the beginning of the end of the confederacy 's bid for independence . now , what you may not know about the battle of gettysburg , is that is was almost the end of the war , in fact , lee took his army , trying to cross back over the potomac into the south , and the potomac was flooded so , he and his army were pretty much pinned , between this flooded river , and the forces of meade in the north . now meade , if he had attacked , probably could 've won the war right there and then , and lincoln was so angry that meade did n't attack , he wrote him this really nasty letter saying , `` i think you do n't even realize what you 've done here by letting lee get away , we could 've ended the war right now '' , but actually lincoln did n't send that letter , he thought better of it , and instead congratulated meade on his great victory and , the boost of morale that it gave the forces of the united states , at gettysburg . so now i 'd like to take some time to talk about the gettysburg address , which is arguably , the most famous speech in american history , it 's pretty up there . and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg . i think the gettysburg address is really interesting , and all of the events surrounding it , the circumstances surrounding it , tell us a lot about the culture and society of the 19th century , the progress of the civil war , and also the way that things are going to kind of be wrapped up in the end of the civil war . what the ultimate message of the war is going to be , and what the blueprint of reuniting the country is going to look like . so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the governor of pennsylvania , contracts out to create a cemetery . and in this period of three and a half months , there are bodies literally rotting on the ground , so it 's a bit of a hellscape , the entire town of gettysburg stinks , they had to burn all the dead horses , so it smells like burning horses , and rotting human flesh . it is not a happy place to be . so the town of gettysburg and the state of pennsylvania , are very eager to get a cemetery underway at gettysburg . and so they begin the process of burying the bodies , and re-burying the bodies , trying to identify the various corpses that are left on the fields . and they ask this man , edward everett , who is really the preeminent orator of his day . he was like the rock concert of the 19th century , to come and give an oration on the dedication of the gettysburg cemetery . and they say , `` everett , do you think you can do this on october 23rd ? '' and everett says , `` no , i definitely wo n't be ready to have a script for an oration by then , so can you push it back to november 19th ? '' so , it 's actually everett who decides , what day the gettysburg address is going to take place on . lincoln , by contrast , was only invited maybe a month or so before , and he was n't really considered the important speaker of the day , that was everett . but lincoln knew , that he wanted to make something of his remarks at gettysburg . now remember that , an election year is coming up in 1864 , it 's been a hard year , gettysburg is the first major victory that the united states forces have had in a long time , so he kinda wants to make sure that he can set the tone of how gettysburg is going to be remembered , and to reconfirm a sense of mission , about the civil war right , when there 's been such a great loss of life , and when you 're standing around looking at that loss of life , it can be very easy to get discouraged and say , okay , maybe we should just end the war , we should have peace now , allow the south to secede and , retain slavery , and lincoln wants to make sure that people come away from this dedication at gettysburg , with a renewed sense of purpose , in continuing to fight the civil war . now there 's a common misconception that , lincoln wrote the gettysburg address on the back of an envelope on the train to gettysburg . that is almost certainly not the case , because lincoln was a planner . remember that he was self-educated , and he always took a lot of time in anything that he wrote . he wrote drafts and got revisions and , wrote yet another draft . he liked to be extremely precise with his language , and you can see that throughout pretty much everything that he 's written , that he is an extremely effective and eloquent writer . and that was n't just because he was an extremely eloquent person , he was . that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day . now edward everett was i think , the undisputed champion of giving speeches in his day . he was an incredible speaker , and everyone who was there , actually agreed that , everett did an incredible job speaking , he spoke for over two hours , and if that sounds like a really long time to us , for the 19th century that was actually pretty appropriate . that 's what people expected out of oratory , in the 19th century . they paid attention , they were riveted by it . it was like going to see a movie or a concert today . so people really wanted to hear everett talk for that long . in fact , they were quite confused , when lincoln did n't talk for longer then just a couple of minutes . a lot of people even were reported to say , `` was that it ? '' so here in the center we have a picture of the day at gettysburg and , we 're pretty sure that this is the only confirmed picture of lincoln at gettysburg . now , he 's kinda small here , but i think this is a really interesting picture , cuz it gives you a sense of what lincoln 's stature was at the time , and also the people that he surrounded himself with . so , this is lincoln here , right here in the center , not wearing a hat , looking down , and then he 's surrounded by the important people of his cabinet , so right here , i 'm pretty sure this is william seward , who was the secretary of state , and over here , these are john hay and john nicolay who were lincoln 's personal secretaries . they went everywhere with him , and this guy up here is a little harder to see , that is edward everett . now imagine what it would have been like , to stand on this field , in this growing cemetery at gettysburg , and listen to edward everett , and abraham lincoln talk about the meaning of the battle around you . now remember , that it 's november , so it 's been three and a half months since the battle , but the battle of gettysburg took place , in the beginning of july , and it was 90 , 100 degrees outside , so when lee and meade left gettysburg , they left 8,000 or more bodies , rotting in the hot july sun . and many of them had been out there rotting , for those three months so , when you were standing on this field at gettysburg , there would have literally been human bones around you , that you could see . it probably would have still smelled pretty terrible , so you 're really kind of in the thick of the destruction of the civil war , and listening to these two men , who are trying to make meaning out of it for you , so everett gets up , and he gives this fiery speech for two hours , and he goes through all of the details of the battle and says , `` this is what happened over on that hill , and this is what happened over on that hill '' , and he tries to rev up the crowd into kind of this patriotic fervor , of not only appreciating the glory of the union victory at gettysburg , but also renewing their hatred for their enemy , and then lincoln gets up to speak , and he speaks for just a couple minutes , and we 'll talk more about that , in the next video .
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and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg .
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kim describes the individuals in the photograph taken at gettysburg on november 19 , 1863. who is the tall , bearded man in the top hat to the left of lincoln ( to the right of the viewer of the photograph ) ?
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alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt at an invasion , and once again , was turned away by the forces of union general , george meade . gettysburg was the most destructive battle of the civil war , there were about 50,000 casualties , and it , along with the victory of the siege of vicksburg , which followed the day after , on july 4th , really start to signify the beginning of the end of the confederacy 's bid for independence . now , what you may not know about the battle of gettysburg , is that is was almost the end of the war , in fact , lee took his army , trying to cross back over the potomac into the south , and the potomac was flooded so , he and his army were pretty much pinned , between this flooded river , and the forces of meade in the north . now meade , if he had attacked , probably could 've won the war right there and then , and lincoln was so angry that meade did n't attack , he wrote him this really nasty letter saying , `` i think you do n't even realize what you 've done here by letting lee get away , we could 've ended the war right now '' , but actually lincoln did n't send that letter , he thought better of it , and instead congratulated meade on his great victory and , the boost of morale that it gave the forces of the united states , at gettysburg . so now i 'd like to take some time to talk about the gettysburg address , which is arguably , the most famous speech in american history , it 's pretty up there . and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg . i think the gettysburg address is really interesting , and all of the events surrounding it , the circumstances surrounding it , tell us a lot about the culture and society of the 19th century , the progress of the civil war , and also the way that things are going to kind of be wrapped up in the end of the civil war . what the ultimate message of the war is going to be , and what the blueprint of reuniting the country is going to look like . so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the governor of pennsylvania , contracts out to create a cemetery . and in this period of three and a half months , there are bodies literally rotting on the ground , so it 's a bit of a hellscape , the entire town of gettysburg stinks , they had to burn all the dead horses , so it smells like burning horses , and rotting human flesh . it is not a happy place to be . so the town of gettysburg and the state of pennsylvania , are very eager to get a cemetery underway at gettysburg . and so they begin the process of burying the bodies , and re-burying the bodies , trying to identify the various corpses that are left on the fields . and they ask this man , edward everett , who is really the preeminent orator of his day . he was like the rock concert of the 19th century , to come and give an oration on the dedication of the gettysburg cemetery . and they say , `` everett , do you think you can do this on october 23rd ? '' and everett says , `` no , i definitely wo n't be ready to have a script for an oration by then , so can you push it back to november 19th ? '' so , it 's actually everett who decides , what day the gettysburg address is going to take place on . lincoln , by contrast , was only invited maybe a month or so before , and he was n't really considered the important speaker of the day , that was everett . but lincoln knew , that he wanted to make something of his remarks at gettysburg . now remember that , an election year is coming up in 1864 , it 's been a hard year , gettysburg is the first major victory that the united states forces have had in a long time , so he kinda wants to make sure that he can set the tone of how gettysburg is going to be remembered , and to reconfirm a sense of mission , about the civil war right , when there 's been such a great loss of life , and when you 're standing around looking at that loss of life , it can be very easy to get discouraged and say , okay , maybe we should just end the war , we should have peace now , allow the south to secede and , retain slavery , and lincoln wants to make sure that people come away from this dedication at gettysburg , with a renewed sense of purpose , in continuing to fight the civil war . now there 's a common misconception that , lincoln wrote the gettysburg address on the back of an envelope on the train to gettysburg . that is almost certainly not the case , because lincoln was a planner . remember that he was self-educated , and he always took a lot of time in anything that he wrote . he wrote drafts and got revisions and , wrote yet another draft . he liked to be extremely precise with his language , and you can see that throughout pretty much everything that he 's written , that he is an extremely effective and eloquent writer . and that was n't just because he was an extremely eloquent person , he was . that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day . now edward everett was i think , the undisputed champion of giving speeches in his day . he was an incredible speaker , and everyone who was there , actually agreed that , everett did an incredible job speaking , he spoke for over two hours , and if that sounds like a really long time to us , for the 19th century that was actually pretty appropriate . that 's what people expected out of oratory , in the 19th century . they paid attention , they were riveted by it . it was like going to see a movie or a concert today . so people really wanted to hear everett talk for that long . in fact , they were quite confused , when lincoln did n't talk for longer then just a couple of minutes . a lot of people even were reported to say , `` was that it ? '' so here in the center we have a picture of the day at gettysburg and , we 're pretty sure that this is the only confirmed picture of lincoln at gettysburg . now , he 's kinda small here , but i think this is a really interesting picture , cuz it gives you a sense of what lincoln 's stature was at the time , and also the people that he surrounded himself with . so , this is lincoln here , right here in the center , not wearing a hat , looking down , and then he 's surrounded by the important people of his cabinet , so right here , i 'm pretty sure this is william seward , who was the secretary of state , and over here , these are john hay and john nicolay who were lincoln 's personal secretaries . they went everywhere with him , and this guy up here is a little harder to see , that is edward everett . now imagine what it would have been like , to stand on this field , in this growing cemetery at gettysburg , and listen to edward everett , and abraham lincoln talk about the meaning of the battle around you . now remember , that it 's november , so it 's been three and a half months since the battle , but the battle of gettysburg took place , in the beginning of july , and it was 90 , 100 degrees outside , so when lee and meade left gettysburg , they left 8,000 or more bodies , rotting in the hot july sun . and many of them had been out there rotting , for those three months so , when you were standing on this field at gettysburg , there would have literally been human bones around you , that you could see . it probably would have still smelled pretty terrible , so you 're really kind of in the thick of the destruction of the civil war , and listening to these two men , who are trying to make meaning out of it for you , so everett gets up , and he gives this fiery speech for two hours , and he goes through all of the details of the battle and says , `` this is what happened over on that hill , and this is what happened over on that hill '' , and he tries to rev up the crowd into kind of this patriotic fervor , of not only appreciating the glory of the union victory at gettysburg , but also renewing their hatred for their enemy , and then lincoln gets up to speak , and he speaks for just a couple minutes , and we 'll talk more about that , in the next video .
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so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the governor of pennsylvania , contracts out to create a cemetery .
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is there something that says somewhere that `` when you join the union , there is no way out , and if you try to we will fight you to the death until you give up and remain part of the union ... '' , perhaps in the constitution or otherwise ?
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alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt at an invasion , and once again , was turned away by the forces of union general , george meade . gettysburg was the most destructive battle of the civil war , there were about 50,000 casualties , and it , along with the victory of the siege of vicksburg , which followed the day after , on july 4th , really start to signify the beginning of the end of the confederacy 's bid for independence . now , what you may not know about the battle of gettysburg , is that is was almost the end of the war , in fact , lee took his army , trying to cross back over the potomac into the south , and the potomac was flooded so , he and his army were pretty much pinned , between this flooded river , and the forces of meade in the north . now meade , if he had attacked , probably could 've won the war right there and then , and lincoln was so angry that meade did n't attack , he wrote him this really nasty letter saying , `` i think you do n't even realize what you 've done here by letting lee get away , we could 've ended the war right now '' , but actually lincoln did n't send that letter , he thought better of it , and instead congratulated meade on his great victory and , the boost of morale that it gave the forces of the united states , at gettysburg . so now i 'd like to take some time to talk about the gettysburg address , which is arguably , the most famous speech in american history , it 's pretty up there . and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg . i think the gettysburg address is really interesting , and all of the events surrounding it , the circumstances surrounding it , tell us a lot about the culture and society of the 19th century , the progress of the civil war , and also the way that things are going to kind of be wrapped up in the end of the civil war . what the ultimate message of the war is going to be , and what the blueprint of reuniting the country is going to look like . so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the governor of pennsylvania , contracts out to create a cemetery . and in this period of three and a half months , there are bodies literally rotting on the ground , so it 's a bit of a hellscape , the entire town of gettysburg stinks , they had to burn all the dead horses , so it smells like burning horses , and rotting human flesh . it is not a happy place to be . so the town of gettysburg and the state of pennsylvania , are very eager to get a cemetery underway at gettysburg . and so they begin the process of burying the bodies , and re-burying the bodies , trying to identify the various corpses that are left on the fields . and they ask this man , edward everett , who is really the preeminent orator of his day . he was like the rock concert of the 19th century , to come and give an oration on the dedication of the gettysburg cemetery . and they say , `` everett , do you think you can do this on october 23rd ? '' and everett says , `` no , i definitely wo n't be ready to have a script for an oration by then , so can you push it back to november 19th ? '' so , it 's actually everett who decides , what day the gettysburg address is going to take place on . lincoln , by contrast , was only invited maybe a month or so before , and he was n't really considered the important speaker of the day , that was everett . but lincoln knew , that he wanted to make something of his remarks at gettysburg . now remember that , an election year is coming up in 1864 , it 's been a hard year , gettysburg is the first major victory that the united states forces have had in a long time , so he kinda wants to make sure that he can set the tone of how gettysburg is going to be remembered , and to reconfirm a sense of mission , about the civil war right , when there 's been such a great loss of life , and when you 're standing around looking at that loss of life , it can be very easy to get discouraged and say , okay , maybe we should just end the war , we should have peace now , allow the south to secede and , retain slavery , and lincoln wants to make sure that people come away from this dedication at gettysburg , with a renewed sense of purpose , in continuing to fight the civil war . now there 's a common misconception that , lincoln wrote the gettysburg address on the back of an envelope on the train to gettysburg . that is almost certainly not the case , because lincoln was a planner . remember that he was self-educated , and he always took a lot of time in anything that he wrote . he wrote drafts and got revisions and , wrote yet another draft . he liked to be extremely precise with his language , and you can see that throughout pretty much everything that he 's written , that he is an extremely effective and eloquent writer . and that was n't just because he was an extremely eloquent person , he was . that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day . now edward everett was i think , the undisputed champion of giving speeches in his day . he was an incredible speaker , and everyone who was there , actually agreed that , everett did an incredible job speaking , he spoke for over two hours , and if that sounds like a really long time to us , for the 19th century that was actually pretty appropriate . that 's what people expected out of oratory , in the 19th century . they paid attention , they were riveted by it . it was like going to see a movie or a concert today . so people really wanted to hear everett talk for that long . in fact , they were quite confused , when lincoln did n't talk for longer then just a couple of minutes . a lot of people even were reported to say , `` was that it ? '' so here in the center we have a picture of the day at gettysburg and , we 're pretty sure that this is the only confirmed picture of lincoln at gettysburg . now , he 's kinda small here , but i think this is a really interesting picture , cuz it gives you a sense of what lincoln 's stature was at the time , and also the people that he surrounded himself with . so , this is lincoln here , right here in the center , not wearing a hat , looking down , and then he 's surrounded by the important people of his cabinet , so right here , i 'm pretty sure this is william seward , who was the secretary of state , and over here , these are john hay and john nicolay who were lincoln 's personal secretaries . they went everywhere with him , and this guy up here is a little harder to see , that is edward everett . now imagine what it would have been like , to stand on this field , in this growing cemetery at gettysburg , and listen to edward everett , and abraham lincoln talk about the meaning of the battle around you . now remember , that it 's november , so it 's been three and a half months since the battle , but the battle of gettysburg took place , in the beginning of july , and it was 90 , 100 degrees outside , so when lee and meade left gettysburg , they left 8,000 or more bodies , rotting in the hot july sun . and many of them had been out there rotting , for those three months so , when you were standing on this field at gettysburg , there would have literally been human bones around you , that you could see . it probably would have still smelled pretty terrible , so you 're really kind of in the thick of the destruction of the civil war , and listening to these two men , who are trying to make meaning out of it for you , so everett gets up , and he gives this fiery speech for two hours , and he goes through all of the details of the battle and says , `` this is what happened over on that hill , and this is what happened over on that hill '' , and he tries to rev up the crowd into kind of this patriotic fervor , of not only appreciating the glory of the union victory at gettysburg , but also renewing their hatred for their enemy , and then lincoln gets up to speak , and he speaks for just a couple minutes , and we 'll talk more about that , in the next video .
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so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him .
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but how did these leaders ( and the other important union and confederate army leaders ) relate in position and time to each other ?
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alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt at an invasion , and once again , was turned away by the forces of union general , george meade . gettysburg was the most destructive battle of the civil war , there were about 50,000 casualties , and it , along with the victory of the siege of vicksburg , which followed the day after , on july 4th , really start to signify the beginning of the end of the confederacy 's bid for independence . now , what you may not know about the battle of gettysburg , is that is was almost the end of the war , in fact , lee took his army , trying to cross back over the potomac into the south , and the potomac was flooded so , he and his army were pretty much pinned , between this flooded river , and the forces of meade in the north . now meade , if he had attacked , probably could 've won the war right there and then , and lincoln was so angry that meade did n't attack , he wrote him this really nasty letter saying , `` i think you do n't even realize what you 've done here by letting lee get away , we could 've ended the war right now '' , but actually lincoln did n't send that letter , he thought better of it , and instead congratulated meade on his great victory and , the boost of morale that it gave the forces of the united states , at gettysburg . so now i 'd like to take some time to talk about the gettysburg address , which is arguably , the most famous speech in american history , it 's pretty up there . and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg . i think the gettysburg address is really interesting , and all of the events surrounding it , the circumstances surrounding it , tell us a lot about the culture and society of the 19th century , the progress of the civil war , and also the way that things are going to kind of be wrapped up in the end of the civil war . what the ultimate message of the war is going to be , and what the blueprint of reuniting the country is going to look like . so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the governor of pennsylvania , contracts out to create a cemetery . and in this period of three and a half months , there are bodies literally rotting on the ground , so it 's a bit of a hellscape , the entire town of gettysburg stinks , they had to burn all the dead horses , so it smells like burning horses , and rotting human flesh . it is not a happy place to be . so the town of gettysburg and the state of pennsylvania , are very eager to get a cemetery underway at gettysburg . and so they begin the process of burying the bodies , and re-burying the bodies , trying to identify the various corpses that are left on the fields . and they ask this man , edward everett , who is really the preeminent orator of his day . he was like the rock concert of the 19th century , to come and give an oration on the dedication of the gettysburg cemetery . and they say , `` everett , do you think you can do this on october 23rd ? '' and everett says , `` no , i definitely wo n't be ready to have a script for an oration by then , so can you push it back to november 19th ? '' so , it 's actually everett who decides , what day the gettysburg address is going to take place on . lincoln , by contrast , was only invited maybe a month or so before , and he was n't really considered the important speaker of the day , that was everett . but lincoln knew , that he wanted to make something of his remarks at gettysburg . now remember that , an election year is coming up in 1864 , it 's been a hard year , gettysburg is the first major victory that the united states forces have had in a long time , so he kinda wants to make sure that he can set the tone of how gettysburg is going to be remembered , and to reconfirm a sense of mission , about the civil war right , when there 's been such a great loss of life , and when you 're standing around looking at that loss of life , it can be very easy to get discouraged and say , okay , maybe we should just end the war , we should have peace now , allow the south to secede and , retain slavery , and lincoln wants to make sure that people come away from this dedication at gettysburg , with a renewed sense of purpose , in continuing to fight the civil war . now there 's a common misconception that , lincoln wrote the gettysburg address on the back of an envelope on the train to gettysburg . that is almost certainly not the case , because lincoln was a planner . remember that he was self-educated , and he always took a lot of time in anything that he wrote . he wrote drafts and got revisions and , wrote yet another draft . he liked to be extremely precise with his language , and you can see that throughout pretty much everything that he 's written , that he is an extremely effective and eloquent writer . and that was n't just because he was an extremely eloquent person , he was . that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day . now edward everett was i think , the undisputed champion of giving speeches in his day . he was an incredible speaker , and everyone who was there , actually agreed that , everett did an incredible job speaking , he spoke for over two hours , and if that sounds like a really long time to us , for the 19th century that was actually pretty appropriate . that 's what people expected out of oratory , in the 19th century . they paid attention , they were riveted by it . it was like going to see a movie or a concert today . so people really wanted to hear everett talk for that long . in fact , they were quite confused , when lincoln did n't talk for longer then just a couple of minutes . a lot of people even were reported to say , `` was that it ? '' so here in the center we have a picture of the day at gettysburg and , we 're pretty sure that this is the only confirmed picture of lincoln at gettysburg . now , he 's kinda small here , but i think this is a really interesting picture , cuz it gives you a sense of what lincoln 's stature was at the time , and also the people that he surrounded himself with . so , this is lincoln here , right here in the center , not wearing a hat , looking down , and then he 's surrounded by the important people of his cabinet , so right here , i 'm pretty sure this is william seward , who was the secretary of state , and over here , these are john hay and john nicolay who were lincoln 's personal secretaries . they went everywhere with him , and this guy up here is a little harder to see , that is edward everett . now imagine what it would have been like , to stand on this field , in this growing cemetery at gettysburg , and listen to edward everett , and abraham lincoln talk about the meaning of the battle around you . now remember , that it 's november , so it 's been three and a half months since the battle , but the battle of gettysburg took place , in the beginning of july , and it was 90 , 100 degrees outside , so when lee and meade left gettysburg , they left 8,000 or more bodies , rotting in the hot july sun . and many of them had been out there rotting , for those three months so , when you were standing on this field at gettysburg , there would have literally been human bones around you , that you could see . it probably would have still smelled pretty terrible , so you 're really kind of in the thick of the destruction of the civil war , and listening to these two men , who are trying to make meaning out of it for you , so everett gets up , and he gives this fiery speech for two hours , and he goes through all of the details of the battle and says , `` this is what happened over on that hill , and this is what happened over on that hill '' , and he tries to rev up the crowd into kind of this patriotic fervor , of not only appreciating the glory of the union victory at gettysburg , but also renewing their hatred for their enemy , and then lincoln gets up to speak , and he speaks for just a couple minutes , and we 'll talk more about that , in the next video .
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that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day .
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were there any slave-owners in the crowd at the time ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again .
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can anybody tell me how many actual grains of rice would be on the last ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth .
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if a tree that splits twice is a binary tree , what is a tree that splits four times ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: well i guess why not ? it 's different . it 's not going to be 2 anymore .
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are exponents the same as indices or is it completely different ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people .
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why when i think of a mount everest sized mound of rice , it feels impossible ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that .
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so the grains of rice and the game broad and the popsicle sticks all are divided by two ... 246810 ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that .
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what is the point of the rice and popsicle sticks ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people .
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is n't mount everest too much for the pile of rice which is 2^63 ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks .
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does any number to the 0 power equal 1 ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks .
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i have to ask is it possible to divide a number by 1 ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree .
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what is four squared to the 3rd power ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice .
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what would be the amount on the first square on the back of the board ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together .
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why does a tree appear ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: hi sal . : hey britt .
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what kind of drawing program does sal use ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: yeah , they have things to do . they have to go home and count grains of rice . : right .
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how many grains of rice be on the chessboard if they finish putting all the rice ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: yeah , they have things to do . they have to go home and count grains of rice . : right .
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how many grains of rice be on the chessboard if they finish putting all the rice ?
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: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i doubled the amount of rice . : mm hm . : how much rice do you think would be on this square ? : on that square ? let me think about it a little bit . actually , i 'm going to take some ... here you have 1 and we multiply that times 2 , so this is going to be 2 times 2 . no , no 2 times 1 , what am i doing ? now this is 2 times that one so this is 2 times 2 . now this is 2 times that . so this is ... okay , we 're starting to take a lot of 2 's here and multiplying them together . so this is 2 times 2 ... i 'm trying to write sideways . times 2 . this one is going to be 5 , 2 's multiplied together . this is going to be 6 , 2 's multiplied together . this is going to be 7 , 2 's multiplied together . 8 , 2 's multiplied together . 9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together . 8,192 grains of rice is what we should see right over here . : and you know , i had fun last night and i was up late , but there you go . : did you really count out 8,192 grains of rice ? : more or less . : okay . let 's just say you did . : what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again . so it 's this number times ... let 's see , 2 times 2 is 4 . times 2 is 8 , times 2 is 16 . so it 's going to get us like 120 , like 130,000 or around there . :131,672 . : you had a lot of time last night . we 're not even halfway across the board yet . : we 're not . : this is a lot of ... that 's a lot of rice , there . you could throw a party . : what about the last square ? this is 63 steps . : we 're going to take 2 times 2 and we 're going to do 63 of those . so this is going to be a huge number . and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people . : but i have one question . i mean , you know , this was a little bit of a pain for me to write all of these 2 's . : so was this . : if i were the mathematical community i would want some type of notation . : you kind of got on it here . i like this dot , dot , dot and the 63 . this i understand this . : yeah , you could understand this but this is still a little bit ... this is a little bit too much . what if , instead , we just wrote ... : mathematicians love being efficient , right ? they 're lazy . : yeah , they have things to do . they have to go home and count grains of rice . : right . ( laughter ) : yeah . so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice . and when we double it we have 2 grains of rice . : yup . : and we double it again we have 4 . i 'm thinking this is similar to what we were doing , it 's just represented differently . : yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that . 1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks . every stage , every branch , you 're multiplying by 2 again . : i basically just continue splitting just like a tree does . : yup . : now i can really see what 2 to the power of 3 looks like . : and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way . : yeah , you could view it ... i guess one way to think about it is how many times you 've branched . but that one , that tree there , is actually even more interesting . : i do n't think this counts because , again , this branches 4 times at each branch . : well i guess why not ? it 's different . it 's not going to be 2 anymore . so the first one where you have n't branched yet , this is going to be 4 to the 0 power . you 've had no branches yet . this , you branched once so now this is 4 to the first power . you have 4 branches now . : oh , i like this . : and now each of those . so now you 've branched twice . so now this is 4 to the second power . so yeah , the base , or what 's called the base when you take an exponent , this 4 right over here . this is how many new branches each of the branches turn into at each of these , i guess , junctions you could say . : let 's call them junctions . : junctions . you have n't branched yet . here you 've branched once , and here you 've branched twice . : this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth . : yes . ( laughs )
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: and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way .
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why could n't of they just stop after sal wrote 2 to the power of 63 ?
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