HMMT 二月 2014 · COMB 赛 · 第 2 题

HMMT February 2014 — COMB Round — Problem 2

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