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Math Prize 2013 Problem 13

Source:

September 10, 2013
probabilityfunctioninequalities

Problem Statement

Each of nn boys and nn girls chooses a random number from the set {1,2,3,4,5}\{ 1, 2, 3, 4, 5 \}, uniformly and independently. Let pnp_n be the probability that every boy chooses a different number than every girl. As nn approaches infinity, what value does pnn\sqrt[n]{p_n} approach?
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