HMMT 十一月 2012 · 团队赛 · 第 5 题

英文解析

5. [ 4 ] Let π be a randomly chosen permutation of the numbers from 1 through 2012. Find the probabilitythat π ( π (2012)) = 2012.
   Answer: There are two possibilities: either π (2012) = 2012 or π (2012) = i and π ( i ) = 2012 for 1
   1006
   i 6 = 2012. The first case occurs with probability 2011! / 2012! = 1 / 2012, since any permutation on theremaining 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. Sincethere are 2011 possible values for i , and since our two possibilities are disjoint, the overall probabilitythat π ( π (2012)) = 2012 equals
   1 1 1 + (2011) = .
   2012 2011 · 2012 1006