# Difference between revisions of "1994 AHSME Problems/Problem 30"

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<math> \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 </math> | <math> \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 </math> | ||

==Solution== | ==Solution== | ||

− | + | Since the sum of dice is 1994, there are at least 333 dice. Now consider the following numbers <math>\underbrace { 6+6+ \dots + 6 }_{ 332 } + 2</math> and <math>\underbrace { 1+1+ \dots + 1 }_{ 332 } + 5</math>. 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 <math>S</math> which is <math>1 \cdot 332 + 5 = 337</math>. Only smaller numbers to consider are <math>1 \cdot 332 + 4</math>, <math>1 \cdot 332 + 3</math>, <math>1 \cdot 332 + 2</math>, <math>1 \cdot 332 + 1</math>. As it turns out, none of these work. |

## Revision as of 01:31, 16 August 2017

## Problem

When 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 . The smallest possible value of is

## Solution

Since the sum of dice is 1994, there are at least 333 dice. Now consider the following numbers and . 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 which is . Only smaller numbers to consider are , , , . As it turns out, none of these work.