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3n students participate in a test

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September 9, 2010
probabilitycombinatoricscountingIMO Shortlist

Problem Statement

In a test, 3n3n students participate, who are located in three rows of nn students in each. The students leave the test room one by one. If N1(t),N2(t),N3(t)N_1(t), N_2(t), N_3(t) denote the numbers of students in the first, second, and third row respectively at time tt, find the probability that for each t during the test, ∣Ni(t)−Nj(t)∣<2,i≠j,i,j=1,2,… .|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .
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