HMMT 二月 2004 · GEN2 赛 · 第 5 题

英文解析

5. Eight strangers are preparing to play bridge. How many ways can they be groupedinto two bridge games — that is, into unordered pairs of unordered pairs of people?
   Solution: 315
   Putting 8 people into 4 pairs and putting those 4 pairs into 2 pairs of pairs are independent. If the people are numbered from 1 to 8, there are 7 ways to choose the personto pair with person 1. Then there are 5 ways to choose the person to pair with theperson who has the lowest remaining number, 3 ways to choose the next, and 1 way tochoose the last (because there are only 2 people remaining). Thus, there are 7 · 5 · 3 · 1
   ways to assign 8 people to pairs and similarly there are 3 · 1 ways to assign 4 pairs to
   2 pairs of pairs, so there are 7 · 5 · 3 · 3 = 315 ways.