HMMT 二月 2014 · COMB 赛 · 第 2 题

HMMT February 2014 — COMB Round — Problem 2

个人笔记

There are 10 people who want to choose a committee of 5 people among them. They do this by firstelecting a set of 1, 2, 3, or 4 committee leaders, who then choose among the remaining people tocomplete the 5-person committee. In how many ways can the committee be formed, assuming thatpeople are distinguishable? (Two committees that have the same members but di ↵ erent sets of leadersare considered to be distinct.)

解析

There are 10 people who want to choose a committee of 5 people among them. They do this by firstelecting a set of 1, 2, 3, or 4 committee leaders, who then choose among the remaining people tocomplete the 5-person committee. In how many ways can the committee be formed, assuming thatpeople are distinguishable? (Two committees that have the same members but different sets of leadersare considered to be distinct.)
( )
Answer: 7560 There are ways to choose the 5-person committee. After choosing the 10
( )5
committee, there are 2 − 2 = 30 ways to choose the leaders. So the answer is 30 · = 7560.510 5