HMMT 十一月 2009 · 冲刺赛 · 第 13 题

HMMT November 2009 — Guts Round — Problem 13

[ 8 ] 8 students are practicing for a math contest, and they divide into pairs to take a practice test. In how many ways can they be split up? 4 3 2

解析

[ 8 ] 8 students are practicing for a math contest, and they divide into pairs to take a practice test. In how many ways can they be split up? Answer: 105 We create the pairs one at a time. The first person has 7 possible partners. Set this pair aside. Of the remaining six people, pick a person. He or she has 5 possible partners. Set this pair aside. Of the remaining four people, pick a person. He or she has 3 possible partners. Set this pair aside. Then the last two must be partners. So there are 7 · 5 · 3 = 105 possible groupings. Alternatively, we can consider the 8! permutations of the students in a line, where the first two are a pair, the next two are a pair, etc. Given a grouping, there are 4! ways to arrange the four pairs in order, and in each 8! pair, 2 ways to order the students. So our answer is = 7 · 5 · 3 = 105. 4 4!2 4 3 2