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Difference between revisions of "Mock AIME II 2012 Problems/Problem 5"

(Created page with "==Problem== A fair die with <math>12</math> sides numbered <math>1</math> through <math>12</math> inclusive is rolled <math>n</math> times. The probability that the sum of the r...")
 
 
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==Solution==
 
==Solution==
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Consider the generating function for a 12 sided die. When rolled n times, the generating function is <math>(x^1+x^2+\hdots+x^12)^n</math>. This polynomial is clearly symmetric, and the coefficient of <math>x^k</math> is thus the same as the coefficient of <math>x^{13n-k}</math>.  
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Consider the generating function for a 12 sided die. When rolled n times, the generating function is <math>(x^1+x^2+\hdots+x^{12})^n</math>. This polynomial is clearly symmetric, and the coefficient of <math>x^k</math> is thus the same as the coefficient of <math>x^{13n-k}</math>.  
  
 
Thus, the coefficient of <math>x^{2012}</math> is the same as the coefficient of <math>x^{13n-2012}</math>. Note that <math>n \leq 2012 \leq 12n</math>, and thus the minimum value for n is 168. The minimum value of <math>13n-741</math> is thus <math>13(168)-2012=172</math>, so the answer is <math>\boxed{172}</math>.
 
Thus, the coefficient of <math>x^{2012}</math> is the same as the coefficient of <math>x^{13n-2012}</math>. Note that <math>n \leq 2012 \leq 12n</math>, and thus the minimum value for n is 168. The minimum value of <math>13n-741</math> is thus <math>13(168)-2012=172</math>, so the answer is <math>\boxed{172}</math>.

Latest revision as of 02:01, 5 April 2012

Problem

A fair die with $12$ sides numbered $1$ through $12$ inclusive is rolled $n$ times. The probability that the sum of the rolls is $2012$ is nonzero and is equivalent to the probability that a sum of $k$ is rolled. Find the minimum value of $k$.

Solution

Consider the generating function for a 12 sided die. When rolled n times, the generating function is $(x^1+x^2+\hdots+x^{12})^n$. This polynomial is clearly symmetric, and the coefficient of $x^k$ is thus the same as the coefficient of $x^{13n-k}$.

Thus, the coefficient of $x^{2012}$ is the same as the coefficient of $x^{13n-2012}$. Note that $n \leq 2012 \leq 12n$, and thus the minimum value for n is 168. The minimum value of $13n-741$ is thus $13(168)-2012=172$, so the answer is $\boxed{172}$.

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