# 1994 AHSME Problems/Problem 30

## Problem

When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is

$\textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341$

## Solution

Since the sum of dice is $1994$, there are at least $333$ dice. Now consider the following numbers $\underbrace { 6+6+ \dots + 6 }_{ 332 } + 2$ and $\underbrace{ 1+1+ \dots + 1 }_{ 332 } + 5$. They both have the same probability of occurring because $6$ negated by switching it to $1$ and $2$ is negated by switching it to $5$. So we have obtained an upper bound over $S$ which is $1 \cdot 332 + 5 = 337$. Only smaller numbers to consider are $1 \cdot 332 + 4$, $1 \cdot 332 + 3$, $1 \cdot 332 + 2$, $1 \cdot 332 + 1$. As it turns out, none of these work.