Mock AIME II 2012 Problems/Problem 5

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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Mock AIME II 2012 Problems/Problem 5

Problem

Solution