Mock AIME II 2012 Problems/Problem 5
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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
.