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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==
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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.
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Since the sum of dice is <math>1994</math>, there are at least <math>333</math> 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 <math>6</math> negated by switching it to <math>1</math> and <math>2</math> is negated by switching it to <math>5</math>. 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 18:55, 12 September 2017

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.

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