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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"If I roll a fair, regular six-sided die four times, what is the probability that I will roll the nu(...TRUNCATED) | "Each roll is independent of every other roll, so the probability of getting a $1$ on any given roll(...TRUNCATED) | Counting & Probability | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number \\(1(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number 1 ex(...TRUNCATED) | true | "To determine the probability of rolling a number \\(1\\) exactly three times when rolling a fair si(...TRUNCATED) | true | 0.181641 | 0.194336 | 0.197266 | 0.206055 | 0.200195 | 3.9375 | 8.25 | 8.5 | 7.46875 | 8.4375 | 0.033203 | 0.984375 | 0.992188 | 0.996094 | 0.996094 | 0.996094 | 1 | 1 | 1 | 1 | -1.539063 | 2.328125 | 2.375 | 2.359375 | 1.773438 | "To determine the probability that you will roll the number \\(1\\) exactly three times when rolling(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times out of four rolls of(...TRUNCATED) | true | "To find the probability that a fair six-sided die rolled four times results in the number 1 exactly(...TRUNCATED) | true | "To find the probability that a fair six-sided die rolled four times results in the number 1 exactly(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times in four rolls of a f(...TRUNCATED) | true | "To find the probability of rolling the number 1 exactly three times in four rolls of a fair six-sid(...TRUNCATED) | true | "To determine the probability that a fair six-sided die rolled four times will result in the number (...TRUNCATED) | true | "To determine the probability of rolling the number 1 exactly three times in four rolls of a fair si(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when rolling a fair (...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times in four rolls of a f(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when rolling a fair (...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting exactly three 1(...TRUNCATED) | true | "To find the probability that I will roll the number \\(1\\) exactly three times when rolling a fair(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when rolling a fair (...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number 1 ex(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number 1 ex(...TRUNCATED) | true | "To determine the probability of rolling a fair, regular six-sided die four times and getting the nu(...TRUNCATED) | true | "To determine the probability that you will roll the number \\(1\\) exactly three times when rolling(...TRUNCATED) | true | "To find the probability of rolling a fair six-sided die four times and getting the number \\(1\\) e(...TRUNCATED) | true | "To find the probability of rolling a fair six-sided die four times and getting exactly three times (...TRUNCATED) | false | "To determine the probability that a fair six-sided die rolled four times will result in the number (...TRUNCATED) | true | "To find the probability of rolling the number \\(1\\) exactly three times when a fair six-sided die(...TRUNCATED) | false | "To find the probability that we will roll the number \\(1\\) exactly three times when rolling a fai(...TRUNCATED) | false | "To find the probability that a fair six-sided die rolled four times will show the number 1 exactly (...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times in four rolls of a f(...TRUNCATED) | false | "To determine the probability of rolling the number \\(1\\) exactly three times when rolling a fair (...TRUNCATED) | true | "To find the probability of rolling a fair six-sided die four times and getting the number 1 exactly(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times in four rolls of a f(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number 1 ex(...TRUNCATED) | true | "To determine the probability of rolling a number \\(1\\) exactly three times when rolling a fair si(...TRUNCATED) | true | "To determine the probability of rolling the number 1 exactly three times in four rolls of a fair si(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when rolling a fair (...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when rolling a fair (...TRUNCATED) | true | "To determine the probability of rolling a number \\(1\\) exactly three times in four rolls of a fai(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting exactly three 1(...TRUNCATED) | true | "To find the probability of rolling a number \\(1\\) exactly three times in four rolls of a fair six(...TRUNCATED) | true | "To determine the probability of rolling a number 1 exactly three times out of four rolls of a fair (...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number 1 ex(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number \\(1(...TRUNCATED) | true | "To find the probability of rolling a fair six-sided die four times and getting the number 1 exactly(...TRUNCATED) | true | "To determine the probability of rolling a number \\(1\\) exactly three times out of four rolls of a(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and rolling the number 1 ex(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number \\(1(...TRUNCATED) | true | "To find the probability of rolling a number \\(1\\) exactly three times in four rolls of a fair six(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and rolling the number 1 ex(...TRUNCATED) | true | "To find the probability that I will roll the number \\(1\\) exactly three times when rolling a fair(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when rolling a fair (...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times in four rolls of a f(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number 1 ex(...TRUNCATED) | false | "To determine the probability that you will roll the number \\(1\\) exactly three times when you rol(...TRUNCATED) | true | "To find the probability of rolling a fair six-sided die four times and getting the number \\(1\\) e(...TRUNCATED) | false | "To determine the probability that a fair six-sided die rolled four times will result in the number (...TRUNCATED) | true | 1 | 2.25 | 2.546875 | 2.390625 | 2.328125 | 1.859375 | 2.375 | 1 | 1 | 1 | 0.996094 | 0.996094 |
"Find the greatest common divisor of $7!$ and $(5!)^2.$\nPlease reason step by step, and put your fi(...TRUNCATED) | "$$ \\begin{array}{rcrcr} 7! &=& 7 \\cdot 6 \\cdot 5 \\cdot 4 \\cdot 3 \\cdot 2 \\cdot 1 &=& 2^4 \\c(...TRUNCATED) | Counting & Probability | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we will first compute the p(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | 0.253906 | 0.251953 | 0.238281 | 0.273438 | 0.271484 | 2.234375 | 6.28125 | 4.59375 | 6.40625 | 5.3125 | 0.949219 | 0.964844 | 1 | 0.996094 | 0.992188 | 0.796875 | 1 | 1 | 1 | 0.996094 | 0.683594 | 3.484375 | 2.765625 | 2.890625 | 2.96875 | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to determine (...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing both(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing thes(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing both(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing thes(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing thes(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing both(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to compute th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we will first express each (...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express ea(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by finding the pri(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start bydateFormatating (...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by expressing each(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\( 7! \\) and \\( (5!)^2 \\), we will first determine(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by determining the(...TRUNCATED) | true | 1 | 3 | 3.296875 | 3.125 | 2.578125 | 3.40625 | 3.28125 | 1 | 1 | 1 | 0.996094 | 0.996094 |
"An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ (...TRUNCATED) | "First, we find the probability both are green, then the probability both are blue, and add the two (...TRUNCATED) | Counting & Probability | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To solve for \\( N \\), we need to determine the probability that both balls drawn from the urns ar(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability of drawing two balls of the same color fro(...TRUNCATED) | true | 0.147461 | 0.167969 | 0.163086 | 0.142578 | 0.12793 | 6.25 | 11.25 | 9.8125 | 8.3125 | 5.84375 | 0.130859 | 0.996094 | 0.992188 | 0.984375 | 0.96875 | 0.394531 | 1 | 0.996094 | 0.996094 | 0.996094 | 0.180664 | 3.6875 | 3.796875 | 4.15625 | 4.25 | "To solve the problem, we need to determine the value of \\( N \\) such that the probability of draw(...TRUNCATED) | true | "To solve the problem, we need to determine the value of \\( N \\) such that the probability of draw(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To find the value of \\(N\\), we need to calculate the probability that both balls drawn from the t(...TRUNCATED) | true | "To find the value of \\( N \\) that satisfies the given probability condition, we start by calculat(...TRUNCATED) | true | "To determine the value of \\( N \\), we need to calculate the probability that both balls drawn (on(...TRUNCATED) | true | "To solve the problem, we need to calculate the probability of drawing two balls of the same color f(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To solve the problem, we need to calculate the probability of drawing two balls of the same color f(...TRUNCATED) | true | "To find the value of \\(N\\) for which the probability that both balls drawn from the two urns are (...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability of drawing two balls of the same color fro(...TRUNCATED) | true | "To determine the value of \\( N \\) such that the probability of drawing two balls of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\), we start by calculating the probability that both balls drawn from (...TRUNCATED) | true | "To solve for \\( N \\), we need to determine the probability that both balls drawn from the two urn(...TRUNCATED) | true | "To determine the value of \\( N \\), we need to calculate the probability of drawing two balls of t(...TRUNCATED) | true | "To determine the value of \\( N \\) such that the probability of drawing two balls of the same colo(...TRUNCATED) | true | "To determine the value of \\(N\\), we need to calculate the probability of drawing two balls of the(...TRUNCATED) | true | "To find the value of \\( N \\), we first need to determine the probability that both balls drawn fr(...TRUNCATED) | true | "To find the value of \\(N\\), we need to calculate the probabilities of drawing balls of the same c(...TRUNCATED) | false | "To find the value of \\(N\\), we start by calculating the probabilities of drawing balls of the sam(...TRUNCATED) | false | "To find the value of \\(N\\) such that the probability of drawing two balls of the same color from (...TRUNCATED) | false | "To solve the problem, we need to find the value of \\(N\\) such that the probability of drawing a b(...TRUNCATED) | true | "To solve for \\( N \\), we need to consider the probabilities of drawing balls of the same color fr(...TRUNCATED) | true | "To solve the problem, we need to calculate the probability of drawing two balls of the same color, (...TRUNCATED) | false | "To find the value of \\(N\\) for which the probability that both balls drawn from the two urns are (...TRUNCATED) | true | "To solve this problem, we need to find the value of \\( N \\) such that the probability of drawing (...TRUNCATED) | false | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability of drawing two balls of the same color (on(...TRUNCATED) | true | "To determine the value of \\( N \\), we need to calculate the probability that both balls drawn fro(...TRUNCATED) | true | "To solve the problem, we need to calculate the probability that both balls drawn from the two urns (...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability of drawing two balls of the same color fro(...TRUNCATED) | true | "To find the value of \\( N \\), we start by calculating the probability of drawing two balls of the(...TRUNCATED) | true | "To find the value of \\( N \\), we start by calculating the probability of drawing two balls of the(...TRUNCATED) | true | "To find the value of \\( N \\), we start by calculating the probability of drawing two balls of the(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To solve for \\( N \\), we start by calculating the probability of drawing two balls of the same co(...TRUNCATED) | true | "To solve for \\( N \\), we need to calculate the probability of drawing balls of the same color fro(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn are of t(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability of drawing two balls of the same color fro(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn from the(...TRUNCATED) | true | "To determine the value of \\( N \\) such that the probability of drawing two balls of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\), we start by determining the probability that both balls drawn are o(...TRUNCATED) | true | "To solve for \\( N \\), we start by calculating the probability of drawing two balls of the same co(...TRUNCATED) | true | "To determine the value of \\( N \\) such that the probability of drawing two balls of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\), we start by calculating the probability that both balls drawn from (...TRUNCATED) | true | "To solve for \\( N \\), we need to determine the probability that both balls drawn from the two urn(...TRUNCATED) | true | "To determine the value of \\( N \\), we need to calculate the probability of drawing two balls of t(...TRUNCATED) | false | "To determine the value of \\(N\\), let's first identify the贡=localhost problem:\nAn urn contains (...TRUNCATED) | false | "To find the value of \\( N \\), we first need to determine the probability that both balls drawn fr(...TRUNCATED) | true | "To determine the value of \\( N \\) that satisfies the given condition, we analyze the problem step(...TRUNCATED) | false | 1 | 3.96875 | 4.28125 | 4.3125 | 4.5625 | 4.03125 | 4.59375 | 0.996094 | 0.996094 | 0.996094 | 0.996094 | 0.996094 |
"Matt will arrange four identical, dotless dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid be(...TRUNCATED) | "The shortest possible path from $A$ to $B$ requires $4$ dominoes, which is all we have, so we must (...TRUNCATED) | Counting & Probability | "To solve the problem of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | false | "To solve the problem, we need to determine the number of distinct ways to arrange four identical do(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | false | 0.123535 | 0.09375 | 0.109863 | 0.137695 | 0.123047 | 2.34375 | -1.132813 | 1.71875 | 7.84375 | 1.59375 | 0.273438 | 0.001785 | 0.018799 | 0.128906 | 0.322266 | 0.012634 | 0.001099 | 0.00103 | 0.002762 | 0.003479 | 2.765625 | -2.890625 | -4.25 | -1.648438 | -2.34375 | "To determine the number of distinct paths from the upper left-hand corner \\(A\\) to the lower righ(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | true | "To determine the number of distinct arrangements of four dominoes that form a path from the upper l(...TRUNCATED) | true | "To determine the number of distinct arrangements of four dominoes that form a path from the upper l(...TRUNCATED) | false | "To solve this problem, we need to determine the number of distinct arrangements of four identical, (...TRUNCATED) | false | "To solve the problem, we need to determine the number of distinct paths that can be formed by arran(...TRUNCATED) | true | "To solve the problem of finding the number of distinct arrangements of four identical, dotless domi(...TRUNCATED) | true | "To solve the problem, we need to count the number of distinct paths from the top-left corner \\( A (...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotting dominoes on a 5 by 4 gr(...TRUNCATED) | false | "To solve the problem, we need to count the number of distinct paths from the top-left corner \\( A (...TRUNCATED) | true | "To determine the number of distinct paths from the upper left-hand corner \\( A \\) to the lower ri(...TRUNCATED) | true | "To determine the number of distinct paths from point \\(A\\) to point \\(B\\) using four dominoes o(...TRUNCATED) | true | "To solve the problem of finding the number of distinct arrangements of four identical, dotless domi(...TRUNCATED) | false | "To solve the problem, we need to determine the number of distinct arrangements of four dominoes tha(...TRUNCATED) | true | "To determine the number of distinct arrangements of four dominoes that form a path from the upper l(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | true | "To solve the problem of finding how many distinct arrangements of four dominoes form a path from th(...TRUNCATED) | true | "To solve the problem, we need to count the number of distinct paths from the top-left corner \\( A (...TRUNCATED) | false | "To solve the problem, we need to determine the number of ways to form a path from point \\(A\\) to (...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | false | "To solve the problem of finding the number of distinct ways to arrange four identical, dotless domi(...TRUNCATED) | false | "To solve the problem of determining the number of distinct arrangements of four dominoes that form (...TRUNCATED) | false | "To solve this problem, we need to count the number of ways to arrange four dominoes on a 5 by 4 gri(...TRUNCATED) | false | "To solve the problem, we need to count the number of distinct paths from the top-left corner \\( A (...TRUNCATED) | false | "To determine the number of distinct paths from the upper left-hand corner \\( A \\) to the lower ri(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | false | "To solve this problem, we need to determine the number of distinct arrangements of four identical, (...TRUNCATED) | true | "To solve the problem, we need to determine the number of distinct paths from the top-left corner \\(...TRUNCATED) | true | "To solve the problem of finding the number of distinct arrangements of four identical, dotless domi(...TRUNCATED) | false | "To solve the problem, we need to count the number of distinct paths from the top-left corner \\( A (...TRUNCATED) | true | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | true | "To solve the problem, we need to find the number of distinct paths from the upper left-hand corner (...TRUNCATED) | true | "To solve the problem of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | true | "To solve the problem of finding the number of distinct arrangements of four identical, dotless domi(...TRUNCATED) | true | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | true | "To solve the problem of finding the number of distinct arrangements of four identical, dotless domi(...TRUNCATED) | false | "To solve the problem of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | true | "To determine the number of distinct arrangements of four identical, dotless dominoes that form a pa(...TRUNCATED) | true | "To solve the problem, we need to count the number of distinct paths from the upper left-hand corner(...TRUNCATED) | false | "To solve the problem of finding the number of distinct arrangements of four identical, dotless domi(...TRUNCATED) | true | "To determine the number of distinct arrangements of four identical, dotless dominoes that form a pa(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes (shaded 1 by 2(...TRUNCATED) | false | "To determine the number of distinct paths from point \\(A\\) to point \\(B\\) using four dominoes o(...TRUNCATED) | false | "To solve the problem, we need to determine the number of distinct arrangements of four identical, d(...TRUNCATED) | true | "To solve the problem, we need to determine how many distinct paths can be formed using four dominoe(...TRUNCATED) | false | "To determine the number of distinct arrangements of four dominoes that form a path from the top-lef(...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical, dotless dominoes on a 5 by 4 gr(...TRUNCATED) | false | "To solve the problem of finding how many distinct arrangements of four dominoes form a path from th(...TRUNCATED) | true | "To solve the problem, we need to count the number of distinct paths from the top-left corner \\( A (...TRUNCATED) | false | "To solve the problem, we need to count the number of ways to arrange four dominoes on a 5 by 4 grid(...TRUNCATED) | true | 0.000481 | -0.835938 | 3.78125 | 3.59375 | -4.625 | 3.6875 | -1.289063 | 0.054932 | 0.523438 | 0.001411 | 0.019531 | 0.003937 |
Compute $\dbinom{8}{2}$.
Please reason step by step, and put your final answer within \boxed{}. | $\dbinom{8}{2} = \dfrac{8!}{2!6!}=\dfrac{8\times 7}{2\times 1}=\boxed{28}.$ | Counting & Probability | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we will use the formula for combinations, which is given by:\n\n\\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | 0.251953 | 0.240234 | 0.231445 | 0.244141 | 0.234375 | 4.375 | 7.5625 | 8.1875 | 7.65625 | 7.65625 | 0.992188 | 1 | 1 | 1 | 0.996094 | 1 | 1 | 1 | 1 | 1 | 0.433594 | 2.5 | 2.171875 | 2.5 | 2.5 | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\d(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} = \\fr(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\d(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we start with the definition of the binomial coefficient:\n\n\\[\n(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\bi(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we will use the formula for combinations:\n\n\\[\n\\dbinom{n}{k} =(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\bi(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\bi(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\d(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we start with the formula for combinations, which is given by:\n\n(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), which represents the number of ways to choose 2 items from 8 items (...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we start by recalling the formula for combinations, which is given(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), which represents the number of combinations of 8 items taken 2 at (...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we start with the formula for combinations, which is given by:\n\n\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\jetaCal{8}{2}\\), we use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} = \\f(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the binomial coefficient formula:\n\n\\[\n\\dbinom{n}{r} = (...TRUNCATED) | false | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} = \\fr(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we will use the formula for combinations:\n\n\\[\n\\dbinom{n}{k} =(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\bi(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\bi(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by\n\n\\[\n\\d(...TRUNCATED) | true | "To compute \\(\\dbinom{8}{2}\\), we start with the formula for combinations:\n\\[\n\\dbinom{n}{r} =(...TRUNCATED) | false | "To compute \\(\\binom{8}{2}\\), we will use the formula for combinations, which is given by:\n\n\\[(...TRUNCATED) | false | "To compute \\(\\dbinom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\b(...TRUNCATED) | true | 1 | 2.484375 | 2.453125 | 2.40625 | 2.515625 | 2.53125 | 2.3125 | 1 | 1 | 1 | 1 | 0.996094 |
"What is the total number of digits used when the first 2002 positive even integers are written?\nPl(...TRUNCATED) | "First, the $2002$th positive even integer is $4004$, so we are looking for the number of digits use(...TRUNCATED) | Counting & Probability | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | 0.144531 | 0.145508 | 0.154297 | 0.15332 | 0.148438 | 2.71875 | 2.90625 | 6.375 | 5.65625 | 6.96875 | 0.953125 | 0.021606 | 0.523438 | 0.902344 | 0.972656 | 0.851563 | 0.00087 | 0.28125 | 0.945313 | 0.988281 | 3.609375 | -4.5625 | -2.140625 | 4.65625 | 5.0625 | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To find the total number of digits used when the first 2002 positive even integers are written, we (...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To find the total number of digits used when the first 2002 positive even integers are written, we (...TRUNCATED) | true | "To find the total number of digits used when the first 2002 positive even integers are written, we (...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are 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written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To find the total number of digits used when the first 2002 positive even integers are written, we (...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To find the total number of digits used when the first 2002 positive even integers are written, we (...TRUNCATED) | true | "To find the total number of digits used when the first 2002 positive even integers are written, we (...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | true | "To determine the total number of digits used when the first 2002 positive even integers are written(...TRUNCATED) | false | 0.000504 | -4.90625 | 4.46875 | 4.90625 | 4.40625 | 3.96875 | 4.71875 | 1 | 1 | 0.996094 | 0.996094 | 0.996094 |
"Two standard 6-sided dice are tossed. What is the probability that the sum of the numbers shown on (...TRUNCATED) | "There are $6\\cdot6=36$ possible outcomes when two dice are tossed. The greatest possible sum is 12(...TRUNCATED) | Counting & Probability | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | 0.157227 | 0.180664 | 0.186523 | 0.18457 | 0.173828 | 5.375 | 8.5625 | 8.8125 | 8.875 | 9.3125 | 0.151367 | 0.984375 | 0.996094 | 0.996094 | 0.988281 | 0.996094 | 1 | 1 | 1 | 0.996094 | 1.203125 | -1.914063 | -2.078125 | -1.875 | -1.570313 | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To find the probability that the sum of the numbers shown on two standard 6-sided dice is a prime n(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To find the probability that the sum of the numbers shown on two standard 6-sided dice is a prime n(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | false | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | false | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true | 1 | -0.925781 | -1.914063 | -2.09375 | -1.804688 | -1.625 | -1.9375 | 1 | 1 | 1 | 1 | 0.996094 |
"How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes ar(...TRUNCATED) | "Since the boxes are indistinguishable, there are 5 different cases for arrangements of the number o(...TRUNCATED) | Counting & Probability | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | 0.15625 | 0.15332 | 0.146484 | 0.171875 | 0.157227 | 2.625 | 0.410156 | 3.9375 | 8.8125 | 5.09375 | 0.318359 | 0.004913 | 0.014954 | 0.773438 | 0.910156 | 0.135742 | 0.001411 | 0.002472 | 0.921875 | 0.9375 | 1.421875 | -2.703125 | -5.15625 | -0.71875 | -1.609375 | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to distribute 5 distinguishable balls into 3 indistinguishable boxe(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To solve the problem of determining the number of ways to put 5 distinguishable balls into 3 indist(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To solve the problem of determining how many ways there are to put 5 distinguishable balls into 3 i(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to distribute 5 distinguishable balls into 3 indistinguishable boxe(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | true | "To solve the problem of distributing 5 distinguishable balls into 3 indistinguishable boxes, we nee(...TRUNCATED) | true | 0.992188 | 0.062012 | -4.15625 | -0.347656 | -0.890625 | 0.237305 | -2.8125 | 0.001068 | 0.6875 | 0.976563 | 0.988281 | 0.002258 |
"A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without (...TRUNCATED) | "There are three ways to draw two blue marbles and a red one: RBB, BRB, and BBR. Since there are no(...TRUNCATED) | Counting & Probability | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | 0.15918 | 0.21582 | 0.213867 | 0.227539 | 0.233398 | 2.71875 | 10.5 | 10.5 | 9.625 | 10.9375 | 0.373047 | 0.996094 | 0.992188 | 0.988281 | 0.976563 | 0.519531 | 1 | 1 | 0.996094 | 0.996094 | -0.699219 | 2.203125 | 2.125 | 2.203125 | 1.90625 | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To solve the problem of finding the probability that one marble is red and two are blue when three (...TRUNCATED) | true | "To determine the probability of selecting one red marble and two blue marbles from a bag containing(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability of selecting one red marble and two blue marbles from a bag containing(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | false | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability that one marble is red and two is blue when selecting three marbles fr(...TRUNCATED) | false | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To solve the problem of finding the probability that one marble is red and two are blue when three (...TRUNCATED) | true | "To determine the probability of selecting one red marble and two blue marbles from a bag containing(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To determine the probability of selecting one red marble and two blue marbles from a bag containing(...TRUNCATED) | false | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To find the probability that one marble is red and two are bluewhen three marbles are selected at r(...TRUNCATED) | true | "To determine the probability that one runner is red and two are blue, we need to calculate the numb(...TRUNCATED) | true | "To determine the probability that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | 0.996094 | 1.945313 | 1.6875 | 1.9375 | 2.140625 | 2.03125 | 2.375 | 1 | 0.996094 | 0.996094 | 0.996094 | 0.996094 |
"Compute $\\dbinom{16}{5}$.\nPlease reason step by step, and put your final answer within \\boxed{}.(...TRUNCATED) | "$\\dbinom{16}{5}=\\dfrac{16\\times 15\\times 14\\times 13\\times 12}{5\\times 4\\times 3\\times 2\\(...TRUNCATED) | Counting & Probability | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | 0.251953 | 0.231445 | 0.232422 | 0.223633 | 0.229492 | 2.65625 | 5.875 | 6.3125 | 5.3125 | 5.96875 | 0.984375 | 1 | 1 | 0.996094 | 0.992188 | 0.957031 | 1 | 1 | 1 | 0.996094 | 0.808594 | 3.765625 | 3.359375 | 3.734375 | 3.984375 | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for binomial coefficients, which is given by:\n(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient, which is given by(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the binomial coefficient formula:\n\n\\[\n\\dbinom{n}{r} =(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} = \\f(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient, which is given by(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the binomial coefficient formula:\n\\[\n\\dbinom{16}{5} = (...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient:\n\n\\[\n\\binom{n(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for the binomial coefficient, which is given b(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient:\n\n\\[\n\\binom{n(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for the binomial coefficient, which is given b(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | false | "To compute \\(\\binom{16}{5}\\), we need to determine the number of ways to choose 5 items from a s(...TRUNCATED) | false | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we start with the definition of the binomial coefficient:\n\n\\[\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient, which is given by(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), which represents the number of ways to choose 5 elements from a s(...TRUNCATED) | false | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), which represents the number of ways to choose 5 items from a set o(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} = \\f(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we start with the formula for combinations, which is given by:\n\\(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient:\n\n\\[\n\\binom{n(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} = \\f(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations, which is given by\n\n\\[\n\\b(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations, which is given by:\n\n\\[\n\(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we will follow these steps:\n\n1. Identify the values of \\(n\\) a(...TRUNCATED) | false | "To compute \\(\\dbinom{16}{5}\\), we use the binomial coefficient formula:\n\\[\n\\dbinom{16}{5} = (...TRUNCATED) | false | "To compute \\(\\binom{16}{5}\\), we will use the binomial coefficient formula:\n\n\\[\n\\binom{16}{(...TRUNCATED) | false | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient:\n\n\\[\n\\binom{n(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for the binomial coefficient, which is given b(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for the binomial coefficient:\n\n\\[\n\\binom{n(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for the binomial coefficient, which is given b(...TRUNCATED) | true | "To compute \\(\\dbinom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} = \\f(...TRUNCATED) | true | 1 | 3.84375 | 3.96875 | 3.953125 | 4.03125 | 3.59375 | 3.703125 | 1 | 1 | 0.996094 | 0.996094 | 0.996094 |
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