PUMaC 2010 · 组合（B 组） · 第 1 题

PUMaC 2010 — Combinatorics (Division B) — Problem 1

The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can’t play any two tiring songs in a row, how many ways can the band play its 8 songs?

解析

The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are difficult to play. If the Band can’t play any two difficult songs in a row, how many ways can the band play its 8 songs? Answer: 14400 Solution: There are 5! = 120 ways to choose an ordering for the songs that are not difficult to play. Then the setlist is ∗ S ∗ S ∗ S ∗ S ∗ S ∗ , where S represents a song that is not difficult to play, and ∗ represents a space in the setlist that can either be left empty, or filled with ( ) 6 one difficult song. There are = 20 ways to choose 3 of these spaces for the difficult songs, 3 and 3! = 6 ways to choose which difficult song to put in each space. Therefore, the answer is 120 · 20 · 6 = 14400 .