HMMT 二月 2007 · COMB 赛 · 第 1 题

HMMT February 2007 — COMB Round — Problem 1

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

解析

[ 3 ] A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Biff and Jacob must serve together or not at all, and Alice and Jane refuse to serve with each other? ( ) 7 Answer: 41 . If Biff and Jacob are on the committee, there are = 35 ways for the other members 3 ( ) 5 to be chosen. Amongst these 35 possibilities, we reject the = 5 choices where both Alice and Jane 1 ( ) 7 are also serving. If Biff and Jacob are not serving, then there are = 21 ways to choose the remaining 5 ( ) 5 5 members. Again, we reject the = 10 instances where Alice and Jane are chosen, so the total is 3 (35 − 5) + (21 − 10) = 41.