HMMT 二月 2004 · COMB 赛 · 第 6 题

英文解析

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