HMMT 十一月 2012 · 团队赛 · 第 5 题

HMMT November 2012 — Team Round — Problem 5

[ 4 ] Let π be a randomly chosen permutation of the numbers from 1 through 2012. Find the probability that π ( π (2012)) = 2012.

解析

[ 4 ] Let π be a randomly chosen permutation of the numbers from 1 through 2012. Find the probability that π ( π (2012)) = 2012. 1 Answer: There are two possibilities: either π (2012) = 2012 or π (2012) = i and π ( i ) = 2012 for 1006 i 6 = 2012. The first case occurs with probability 2011! / 2012! = 1 / 2012, since any permutation on the remaining 2011 elements is possible. Similarly, for any fixed i , the second case occurs with probability 2010! / 2012! = 1 / (2011 · 2012), since any permutation on the remaining 2010 elements is possible. Since there are 2011 possible values for i , and since our two possibilities are disjoint, the overall probability that π ( π (2012)) = 2012 equals 1 1 1