HMMT 二月 2004 · COMB 赛 · 第 6 题

HMMT February 2004 — COMB Round — Problem 6

A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other?

解析

A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other? Solution: 41 ( ) 7 If Bill and Karl are on the committee, there are = 35 ways for the other group 3 members to be chosen. However, if Alice and Jane are on the committee with Bill and ( ) 5 Karl, there are = 5 ways for the last member to be chosen, yielding 5 unacceptable 1 ( ) 7 committees. If Bill and Karl are not on the committee, there are = 21 ways for 5 the 5 members to be chosen, but again if Alice and Jane were to be on the committee, ( ) 5 there would be = 10 ways to choose the other three members, yielding 10 more 3 unacceptable committees. So, we obtain (35 − 5) + (21 − 10) = 41 ways the committee can be chosen.