Difference between revisions of "Mock AIME II 2012 Problems/Problem 5"
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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>. | + | 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 03:01, 5 April 2012
Problem
A fair die with sides numbered
through
inclusive is rolled
times. The probability that the sum of the rolls is
is nonzero and is equivalent to the probability that a sum of
is rolled. Find the minimum value of
.
Solution
Consider the generating function for a 12 sided die. When rolled n times, the generating function is . This polynomial is clearly symmetric, and the coefficient of
is thus the same as the coefficient of
.
Thus, the coefficient of is the same as the coefficient of
. Note that
, and thus the minimum value for n is 168. The minimum value of
is thus
, so the answer is
.