Difference between revisions of "2005 AMC 10B Problems/Problem 12"
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There are three cases where the product of the numbers is prime. One die will show <math>2</math>, <math>3</math>, or <math>5</math> and each of the other <math>11</math> dice will show a <math>1</math>. For each of these three cases, the number of ways to order the numbers is <math>\dbinom{12}{1}</math> = <math>12</math> . There are <math>6</math> possible numbers for each of the <math>12</math> dice, so the total number of permutations is <math>6^{12}</math>. The probability the product is prime is therefore <math> \frac{3\cdot 12}{6^{12} = \mathrm{(E)} \left(\frac{1}{6}\right)^{10}</math> | There are three cases where the product of the numbers is prime. One die will show <math>2</math>, <math>3</math>, or <math>5</math> and each of the other <math>11</math> dice will show a <math>1</math>. For each of these three cases, the number of ways to order the numbers is <math>\dbinom{12}{1}</math> = <math>12</math> . There are <math>6</math> possible numbers for each of the <math>12</math> dice, so the total number of permutations is <math>6^{12}</math>. The probability the product is prime is therefore <math> \frac{3\cdot 12}{6^{12} = \mathrm{(E)} \left(\frac{1}{6}\right)^{10}</math> | ||
− | + | \left(\dfrac{1}{6}\right)^{11}\times6=\boxed{\mathrm{(E)}\ \left(\dfrac{1}{6}\right)^{10}}<math> | |
− | <math>\frac{3\cdot 12}{6^{12}} | + | </math>\frac{3\cdot 12}{6^{12}}$. |
== See Also == | == See Also == | ||
{{AMC10 box|year=2005|ab=B|num-b=11|num-a=13}} | {{AMC10 box|year=2005|ab=B|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 07:54, 1 June 2021
Contents
Problem
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
Solution
In order for the product of the numbers to be prime, of the dice have to be a , and the other die has to be a prime number. There are prime numbers (, , and ), and there is only one , and there are ways to choose which die will have the prime number, so the probability is .
Solution 2
There are three cases where the product of the numbers is prime. One die will show , , or and each of the other dice will show a . For each of these three cases, the number of ways to order the numbers is = . There are possible numbers for each of the dice, so the total number of permutations is . The probability the product is prime is therefore $\frac{3\cdot 12}{6^{12} = \mathrm{(E)} \left(\frac{1}{6}\right)^{10}$ (Error compiling LaTeX. Unknown error_msg)
\left(\dfrac{1}{6}\right)^{11}\times6=\boxed{\mathrm{(E)}\ \left(\dfrac{1}{6}\right)^{10}}$$ (Error compiling LaTeX. Unknown error_msg)\frac{3\cdot 12}{6^{12}}$.
See Also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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