HMMT 十一月 2009 · 冲刺赛 · 第 13 题

英文解析

13. [ 8 ] 8 students are practicing for a math contest, and they divide into pairs to take a practice test. Inhow 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 thispair aside. Of the remaining six people, pick a person. He or she has 5 possible partners. Set this pairaside. Of the remaining four people, pick a person. He or she has 3 possible partners. Set this pairaside. 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 nexttwo 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!24
    4 3 2