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classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed6 stringlengths 208 9.82k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed6 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed6 stringlengths 60 10.7k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed6 bool 2
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classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed7 stringlengths 98 9.1k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed7 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed7 stringlengths 102 10.1k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed7 bool 2
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classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed8 stringlengths 201 9.04k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed8 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed8 stringlengths 140 8.95k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed8 bool 2
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classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed9 stringlengths 157 9.81k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed9 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed9 stringlengths 122 8.55k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed9 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.3_seed10 stringlengths 246 9.44k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.3_seed10 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed10 stringlengths 135 9.11k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed10 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed10 stringlengths 165 7.6k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed10 bool 2
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classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed11 stringlengths 246 11.9k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed11 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed11 stringlengths 76 7.08k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed11 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.3_seed12 stringlengths 246 10.5k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.3_seed12 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed12 stringlengths 207 10.1k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed12 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed12 stringlengths 71 6.9k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed12 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.3_seed13 stringlengths 247 9.63k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.3_seed13 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed13 stringlengths 239 9.17k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed13 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed13 stringlengths 13 9.43k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed13 bool 2
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classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed14 stringlengths 223 9.14k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed14 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed14 stringlengths 208 9.49k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed14 bool 2
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classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed15 stringlengths 249 9.38k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.5_seed15 bool 2
classes | response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed15 stringlengths 127 8.01k | answer_is_correct_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed15 bool 2
classes | score_gt_output@Qwen2.5MathPRM72B float64 0 1 | score_response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed0@Qwen2.5MathPRM72B float64 0 1 | score_response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed1@Qwen2.5MathPRM72B float64 0 1 | score_response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed2@Qwen2.5MathPRM72B float64 0 1 | score_response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed3@Qwen2.5MathPRM72B float64 0 1 | score_response_Qwen2.5-1.5B-Instruct_p1.0_t0.7_seed4@Qwen2.5MathPRM72B float64 0 1 | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0 stringlengths 201 9.81k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt2 stringlengths 136 10.5k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt2 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt3 stringlengths 24 9.09k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt3 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt4 stringlengths 9 10.4k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt4 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt5 stringlengths 16 9.64k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt5 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt6 stringlengths 3 11k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt6 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt7 stringlengths 3 9.61k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt7 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt8 stringlengths 3 9.59k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt8 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt9 stringlengths 3 8.91k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt9 bool 2
classes | response_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt1 stringlengths 201 9.81k | answer_is_correct_paraphrase_Qwen2.5-1.5B-Instruct_p1.0_t0.0_seed0_prompt1 bool 2
classes |
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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) | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...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 in four rolls of a fair si(...TRUNCATED) | true | "To determine the probability of rolling the number 1 exactly three times in four rolls of a fair, s(...TRUNCATED) | false | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | false | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | false | "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 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 solve this problem, we need to determine the probability of rolling a fair six-sided die four ti(...TRUNCATED) | false | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...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 in four rolls of a f(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...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 in four rolls of a f(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability of rolling the number $1$ exactly three times when a fair, regular six(...TRUNCATED) | false | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability that a fair six-sided die will roll the number 1 exactly three times o(...TRUNCATED) | true | "To determine the probability of rolling the number 1 exactly three times when a fair, six-sided die(...TRUNCATED) | false | "To determine the probability of rolling the number 1 exactly three times when a fair, six-sided die(...TRUNCATED) | false | "To determine the probability of rolling the number 1 exactly three times when a fair, six-sided die(...TRUNCATED) | false | "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 out of four rolls of a fai(...TRUNCATED) | false | "To determine the probability of rolling the number 1 exactly three times out of four rolls of a fai(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | false | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...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 a fair six-sided die (...TRUNCATED) | true | "To determine the probability of rolling the number 1 exactly three times when a fair six-sided die (...TRUNCATED) | true | "To determine the probability of rolling the number 1 exactly three times when a fair six-sided die (...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting exactly three 1(...TRUNCATED) | false | "To determine the probability of rolling a fair six-sided die four times and getting exactly three 1(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting exactly three 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 fair six-sided die four times and getting the number \\(1(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die exactly three times and getting a $1$,(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair six-side(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair six-side(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair six-side(...TRUNCATED) | true | "To determine the probability of rolling the number \\(1\\) exactly three times when a fair, regular(...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 exactly three 1(...TRUNCATED) | true | 0.996094 | 0.001266 | 0.96875 | 0.000767 | 0.914063 | 0.972656 | "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 other than 1 exactly three times in four rolls of (...TRUNCATED) | false | "The probability of rolling a 1 on any given roll is $\\frac{1}{6}$, and the probability of not roll(...TRUNCATED) | true | "The probability of rolling a 1 on any given roll is $\\frac{1}{6}$, and the probability of not roll(...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 six-sided die four times and getting the number 1 ex(...TRUNCATED) | true | "The probability of rolling a $1$ exactly three times when rolling a fair, regular six-sided die fou(...TRUNCATED) | true | "To solve the problem of finding the probability of rolling a number \\(1\\) exactly three times whe(...TRUNCATED) | true | "To determine the probability of rolling a fair six-sided die four times and getting the number 1 ex(...TRUNCATED) | true |
"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) | "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 start by calculating the(...TRUNCATED) | false | "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 start by calculating the(...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 start by calculating the(...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 start by calculating the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we will follow these steps:(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we will follow these steps:(...TRUNCATED) | false | "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 first need to express th(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | false | "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 start by calculating the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | false | "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 start by calculating the(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | false | "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 first need to compute th(...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 calculating the(...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 start by calculating the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating the(...TRUNCATED) | false | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we start by calculating \\((...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 first need to express th(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we first need to express th(...TRUNCATED) | false | "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 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 calculating the(...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 start by calculating the(...TRUNCATED) | false | "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 start by calculating 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 calculating the(...TRUNCATED) | false | "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 first need to compute th(...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 will follow these steps:(...TRUNCATED) | true | "To find the greatest common divisor (GCD) of \\(7!\\) and \\((5!)^2\\), we will follow these steps:(...TRUNCATED) | true | 0.796875 | 0.996094 | 1 | 0.000912 | 0.001457 | 0.005463 | "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 thes(...TRUNCATED) | true | "The greatest common divisor of $7!$ and $(5!)^2$ is found by first expressing each factorial in ter(...TRUNCATED) | true | "The greatest common divisor of $7!$ and $(5!)^2$ is found by first expressing each factorial in ter(...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 thes(...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 thes(...TRUNCATED) | true | "To find the greatest common divisor (gcd) of \\(7!\\) and \\((5!)^2\\), we will follow the steps ou(...TRUNCATED) | true | "To find the greatest common divisor (gcd) of \\(7!\\) and \\((5!)^2\\), we start by expressing thes(...TRUNCATED) | true |
"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) | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn from the two urns ar(...TRUNCATED) | false | "To solve for \\( N \\), we need to calculate the probability that both balls drawn are of the same (...TRUNCATED) | true | "To solve for \\( N \\), we need to calculate the probability that both balls drawn are of the same (...TRUNCATED) | true | "To solve for \\( N \\), we start by determining the probability that both balls drawn are of the sa(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To determine the value of \\( N \\), we start by calculating the probability that both balls drawn (...TRUNCATED) | true | "To determine the value of \\( N \\), we start by calculating the probability that both balls drawn (...TRUNCATED) | true | "To determine the value of \\( N \\) such that the probability that both balls drawn are of the same(...TRUNCATED) | true | "To determine the value of \\( N \\) such that the probability that both balls drawn are of the same(...TRUNCATED) | false | "To determine the value of \\( N \\) such that the probability of drawing two balls of the same colo(...TRUNCATED) | true | "To solve for \\( N \\), we need to calculate the probability that both balls drawn are of the same (...TRUNCATED) | true | "To solve for \\( N \\), we need to calculate the probability that both balls drawn are of the same (...TRUNCATED) | true | "To solve the problem, we need to find the value of \\( N \\) such that the probability that both ba(...TRUNCATED) | false | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn are of t(...TRUNCATED) | true | "To determine the value of \\( N \\), we start by calculating the probability that both balls drawn (...TRUNCATED) | true | "To determine the value of \\( N \\), we start by calculating the probability that both balls drawn (...TRUNCATED) | true | "To determine the value of \\( N \\), we need to calculate the probability that both balls drawn fro(...TRUNCATED) | false | "To determine the value of \\( N \\), we need to calculate the probability that both balls drawn fro(...TRUNCATED) | true | "To determine the value of \\( N \\), we need to calculate the probability that both balls drawn fro(...TRUNCATED) | true | "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 calculate the probability that both balls drawn are of the same (...TRUNCATED) | true | "To determine the value of \\( N \\), we start by calculating the probability that both balls drawn (...TRUNCATED) | true | "To solve for \\( N \\), we need to calculate the probability that both balls drawn are of the same (...TRUNCATED) | true | "To solve for \\( N \\), we need to determine the probability that both balls drawn from the two urn(...TRUNCATED) | false | "To solve for \\( N \\), we need to determine the probability that both balls drawn are of the same (...TRUNCATED) | false | "To find the value of \\( N \\), we start by calculating the probability that both balls drawn are o(...TRUNCATED) | true | "To find the value of \\( N \\), we start by calculating the probability that both balls drawn are o(...TRUNCATED) | true | "To find the value of \\( N \\), we start by calculating the probability that both balls drawn are o(...TRUNCATED) | false | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn from the two urns ar(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn from the two urns ar(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn from the two urns ar(...TRUNCATED) | false | "To find the value of \\( N \\) such that the probability that both balls drawn from the two urns ar(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn from the two urns ar(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability of drawing two balls of the same color is (...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | true | "To determine the value of \\( N \\), we need to calculate the probability that both balls drawn are(...TRUNCATED) | true | "To find the value of \\( N \\), we need to calculate the probability that both balls drawn are of t(...TRUNCATED) | false | "To solve for \\( N \\) in the given problem, we need to calculate the probability that both balls d(...TRUNCATED) | true | "To solve for \\( N \\) given the conditions, let's follow the reasoning step by step.\n\n1. **Ident(...TRUNCATED) | false | "To find the value of \\( N \\) such that the probability that both balls drawn are of the same colo(...TRUNCATED) | false | "To find the value of \\( N \\), we need to determine the probability that both balls drawn from the(...TRUNCATED) | true | "To find the value of \\( N \\), we need to determine the probability that both balls drawn from the(...TRUNCATED) | true | 0.394531 | 0.001152 | 0.992188 | 0.996094 | 0.996094 | 0.002594 | "To solve the problem, we need to find the value of \\( N \\) such that the probability that both ba(...TRUNCATED) | true | "To solve the problem, we need to find the value of \\( N \\) such that the probability that both ba(...TRUNCATED) | true | "The probability that both balls are green is $\\frac{4}{10}\\cdot\\frac{16}{16+N}$, and the probabi(...TRUNCATED) | true | "The probability that both balls are green is $\\frac{4}{10}\\cdot\\frac{16}{16+N}$, and the probabi(...TRUNCATED) | true | "To solve for \\( N \\) given the probabilities of drawing balls of the same color from two urns, we(...TRUNCATED) | true | "To solve for \\( N \\) given the probability that both balls drawn are of the same color is \\( 0.5(...TRUNCATED) | true | "To solve for \\( N \\) in the given problem, we need to find the probability that both balls drawn (...TRUNCATED) | true | \boxed{144} | true | "To solve for \\( N \\) given the probability that both balls drawn are of the same color is \\( 0.5(...TRUNCATED) | true | "To solve the problem, we need to find the value of \\( N \\) such that the probability that both ba(...TRUNCATED) | true |
"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) | "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 of arranging four identical, dotless dominoes on a 5 by 4 grid such that a pat(...TRUNCATED) | false | "To solve this problem, we need to count the number of distinct ways to arrange four dominoes on a 5(...TRUNCATED) | false | "To determine the number of distinct arrangements of four dominoes on a 5 by 4 grid from the upper l(...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve this problem, we need to count the number of distinct ways to arrange four dominoes on a 5(...TRUNCATED) | false | "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 of arranging four identical, dotless dominoes on a 5 by 4 grid from the upper (...TRUNCATED) | false | "To solve the problem of arranging four identical, dotless dominoes on a 5 by 4 grid from the upper (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid from the upper left corn(...TRUNCATED) | false | "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 of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...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 of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "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 of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | false | "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 of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To determine the number of distinct arrangements of four identical dominoes on a 5 by 4 grid such t(...TRUNCATED) | false | "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 of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | false | "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 of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | false | "To solve the problem of arranging four dominoes on a 5 by 4 grid to form a path from the upper left(...TRUNCATED) | false | "To solve the problem of placing dominoes on a 5 by 4 grid to form a path from the top-left corner \(...TRUNCATED) | false | "To solve the problem of arranging four dominoes on a 5 by 4 grid to form a path from the upper left(...TRUNCATED) | false | "To solve the problem of arranging four dominoes on a 5 by 4 grid to form a path from the upper left(...TRUNCATED) | false | "To solve the problem of arranging four dominoes on a 5 by 4 grid to form a path from the upper left(...TRUNCATED) | false | "To solve the problem of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | false | "To determine the number of distinct arrangements of four dominoes on a 5 by 4 grid from the upper l(...TRUNCATED) | false | "To determine the number of distinct arrangements of four dominoes on a 5 by 4 grid from the upper l(...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four dominoes on a 5 by 4 grid to form a path from the top-left c(...TRUNCATED) | false | "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 of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | false | "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 of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | false | "To solve the problem of arranging four dominoes on a 5 by 4 grid to form a path from the top-left c(...TRUNCATED) | false | "To solve the problem of arranging four dominoes on a 5 by 4 grid to form a path from the top-left c(...TRUNCATED) | false | 0.011841 | 0.000357 | 0.001839 | 0.005463 | 0.007935 | 0.018066 | "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 arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | true | "The shortest possible path from $A$ to $B$ requires 4 dominoes, which is all we have, so we must us(...TRUNCATED) | true | "The shortest possible path from $A$ to $B$ requires $4$ dominoes, which is all we have, so we must (...TRUNCATED) | true | "To determine the number of distinct arrangements of four dominoes on a 5 by 4 grid from the upper l(...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 determine the number of distinct arrangements of four dominoes on a 5 by 4 grid from the upper l(...TRUNCATED) | false | "To determine the number of distinct arrangements of four dominoes on a 5 by 4 grid from the upper l(...TRUNCATED) | false | "To solve the problem of arranging four identical dominoes on a 5 by 4 grid to form a path from the (...TRUNCATED) | true | "To solve the problem of arranging four identical, dotless dominoes on a 5 by 4 grid to form a path (...TRUNCATED) | true |
Compute $\dbinom{8}{2}$.
Please reason step by step, and put your final answer within \boxed{}. | $\dbinom{8}{2} = \dfrac{8!}{2!6!}=\dfrac{8\times 7}{2\times 1}=\boxed{28}.$ | "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 \\(\\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 \\(\\binom{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 \\(\\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 will use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \(...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 \\(\\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 \\(\\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:\n\n\\[\n\\binom{n}{r} = \\frac(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\frac(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\frac(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\frac(...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 \\(\\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 \\(\\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 \\(\\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 \\(\\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 by using the definition of the binomial coefficient:\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 \\(\\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 \\(\\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 | 1 | 1 | 1 | 1 | 0.996094 | 1 | "To compute \\(\\binom{8}{2}\\), we start with the formula for combinations:\n\n\\[\n\\binom{n}{r} =(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we start with the formula for combinations:\n\n\\[\n\\binom{n}{r} =(...TRUNCATED) | true | "The binomial coefficient $\\binom{8}{2}$ represents the number of ways to choose 2 items from a set(...TRUNCATED) | true | "The binomial coefficient $\\binom{8}{2}$ represents the number of ways to choose 2 items from a set(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\frac(...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:\n\n\\[\n\\binom{n}{r} = \\frac(...TRUNCATED) | true | \boxed{28} | true | "To compute $\\dbinom{8}{2}$, we can use the formula $\\dbinom{n}{r} = \\dfrac{n!}{r!(n-r)!}$, where(...TRUNCATED) | true | "To compute \\(\\binom{8}{2}\\), we start with the formula for combinations:\n\n\\[\n\\binom{n}{r} =(...TRUNCATED) | true |
"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) | "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) | 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) | 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) | 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) | 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) | 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) | 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) | 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) | false | 0.855469 | 0.000618 | 0.026123 | 0.008057 | 0.011353 | 0.006012 | "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 | "The first 2002 positive even integers are written. The 2002nd positive even integer is 4004. We nee(...TRUNCATED) | true | "The first 2002 positive even integers are written, and we need to find the total number of digits u(...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) | false | The total number of digits used when the first 2002 positive even integers are written is 7456. | 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 |
"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) | "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) | 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) | 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 6-sided dice is a prime numbe(...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 | "First, we need to determine the total number of possible outcomes when two 6-sided dice are tossed.(...TRUNCATED) | true | "First, we need to determine the total number of possible outcomes when two 6-sided dice are tossed.(...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 | 0.996094 | 1 | 1 | 1 | 0.000374 | 0.78125 | "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 | "The probability that the sum of the numbers shown on the dice is a prime number is calculated as fo(...TRUNCATED) | true | "The probability that the sum of the numbers shown on two standard 6-sided dice is a prime number is(...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 | \boxed{\frac{5}{12}} | true | \boxed{\frac{5}{12}} | true | "To solve the problem of finding the probability that the sum of the numbers shown on two standard 6(...TRUNCATED) | true | "To determine the probability that the sum of the numbers shown on two standard 6-sided dice is a pr(...TRUNCATED) | true |
"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) | "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) | false | "To solve the problem of determining how many ways there are to put 5 distinguishable balls into 3 d(...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 distinguishable boxes, we nee(...TRUNCATED) | false | "To determine the number of ways to put 5 distinguishable balls into 3 distinguishable boxes, we nee(...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) | 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) | 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) | 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) | 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) | 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) | 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) | 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) | 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) | 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) | 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 solve the problem of finding the number of ways to put 5 distinguishable balls into 3 indistingu(...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) | 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) | 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) | false | 0.133789 | 0.019531 | 0.332031 | 0.003082 | 0.003372 | 0.00383 | "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 putting 5 distinguishable balls into 3 indistinguishable boxes, we need to (...TRUNCATED) | false | "To solve the problem of distributing 5 distinguishable balls into 3 indistinguishable boxes, we can(...TRUNCATED) | false | "To solve the problem of distributing 5 distinguishable balls into 3 indistinguishable boxes, we nee(...TRUNCATED) | true | "To solve the problem of distributing 5 distinguishable balls into 3 indistinguishable boxes, we nee(...TRUNCATED) | true | "To solve the problem of distributing 5 distinguishable balls into 3 indistinguishable boxes, we nee(...TRUNCATED) | true | "To solve the problem of distributing 5 distinguishable balls into 3 indistinguishable boxes, we nee(...TRUNCATED) | true | "To solve the problem of distributing 5 distinguishable balls into 3 indistinguishable boxes, we nee(...TRUNCATED) | true | "To determine the number of ways to put 5 distinguishable balls into 3 indistinguishable boxes, we n(...TRUNCATED) | false |
"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) | "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 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 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 fro(...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 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 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 solve the problem of finding the probability that one marble is red and two are blue when three (...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 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 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) | false | "To find the probability that one marble is red and two are blue when selecting three marbles at ran(...TRUNCATED) | true | 0.519531 | 1 | 1 | 0.996094 | 0.996094 | 0.996094 | "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 that one marble is red and two are blue when three marbles are selecte(...TRUNCATED) | true | "The probability of drawing one red marble and two blue marbles can be calculated by considering the(...TRUNCATED) | true | "The probability of drawing one red marble and two blue marbles can be calculated by considering the(...TRUNCATED) | true | "To find the probability that one marble is red and two are blue when three marbles are selected at (...TRUNCATED) | true | "To solve the problem of finding the probability that one marble is red and two are blue when three (...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 solve the problem of finding the probability that one marble is red and two are blue when three (...TRUNCATED) | true | "To solve the problem of finding the probability that one marble is red and two are blue when three (...TRUNCATED) | true |
"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) | "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:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...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 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 \\(\\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 \\(\\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 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:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...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 use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{k} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{k} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{k} = \\fra(...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:\n\n\\[\n\\binom{n}{k(...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 combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...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 will use the formula for combinations:\n\n\\[\n\\binom{n}{r} = (...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 combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...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:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | 0.957031 | 1 | 1 | 0.996094 | 0.996094 | 0.996094 | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we start with the formula for combinations:\n\n\\[\n\\binom{n}{k} (...TRUNCATED) | true | "The binomial coefficient $\\binom{16}{5}$ represents the number of ways to choose 5 items from a se(...TRUNCATED) | true | "The binomial coefficient $\\binom{16}{5}$ can be calculated as $\\frac{16 \\times 15 \\times 14 \\t(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...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 can use the formula for combinations:\n\n\\[\n\\dbinom{n}{r} =(...TRUNCATED) | true | "To compute \\(\\binom{16}{5}\\), we use the formula for combinations:\n\n\\[\n\\binom{n}{r} = \\fra(...TRUNCATED) | true |
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