HMMT 二月 2000 · POW 赛 · 第 1 题

英文原题

1. A derangemen t of a string of distin t elemen ts is a rearrangemen t of the string suchthat no elemen t app ears in its original p osition. F or example, BCA is a derangemen t of
   ABC. D represen ts the number of derangemen ts of an y string omp osed of n distin tnelemen ts. D = 1 and D = 2.
   2 3
   (a) What are D and D ?
   4 5
   (b) Ho w man y derangmen ts are there of the string ABCDEF G?
   ( ) Find a re ursiv e relationship for D in terms of the previous t w o terms ( D andn n 1
   D ).
   n 2
   (d) Find a re ursiv e relationship for D in terms of only the previous term, D .
   n n 1