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n students picking n cards with numbers 1-n, probanility to have same no

Source: KJMO 2012 p8

May 4, 2019
probabilitycombinatorics

Problem Statement

Let there be nn students, numbered 11 through nn. Let there be nn cards with numbers 11 through nn written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be pnp_n. Prove that pn−pn−1=0p_n - p_{n-1}=0 if nn is odd, and prove that pn−pn−1=1(−2)kk1−kp_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}} if n=2kn = 2k.
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