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YuchenLi01
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MATH_train_generations

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train · 7.5k rows

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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
"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	0.800781	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
"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	0.382813	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
"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	0.011841	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
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
"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	0.855469	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 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)	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 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)	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)	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 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 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
"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
"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	0.12793	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
"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	0.546875	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
"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	0.960938	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

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