# 1974 AHSME Problems/Problem 24

## Problem

A fair die is rolled six times. The probability of rolling at least a five at least five times is $\mathrm{(A)\ } \frac{13}{729} \qquad \mathrm{(B) \ }\frac{12}{729} \qquad \mathrm{(C) \ } \frac{2}{729} \qquad \mathrm{(D) \ } \frac{3}{729} \qquad \mathrm{(E) \ }\text{none of these}$

## Solution

The probability of rolling at least a five on any one roll is $\frac{2}{6}=\frac{1}{3}$. If there are exactly $5$ fives or sixes rolled, there are $\binom{6}{5}=6$ ways to pick which of the rolls are the fives and sixes, and so the probability in this case is $6\left(\frac{1}{3}\right)^5\left(\frac{2}{3}\right)=\frac{12}{729}$. If there are exactly $6$ fives or sixes rolled, then there is only one way to pick which of the rolls are fives and sixes, so the probability in this case is $\left(\frac{1}{3}\right)^6=\frac{1}{729}$.

Therefore, the total probability is $\frac{12}{729}+\frac{1}{729}=\frac{13}{729}, \boxed{\text{A}}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 