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 00:31, 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.
