HMMT 二月 2008 · COMB 赛 · 第 3 题

HMMT February 2008 — COMB Round — Problem 3

[ 4 ] Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)

解析

[ 4 ] Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other. (Please write your answer as an integer, without any incomplete computations.) Answer: 100800 Since there are 9 cow and pigs combined and 7 horses, there must be a pair with 1 cow and 1 pig, and all the other pairs must contain a horse. There are 4 × 5 ways of selecting the cow-pig pair, and 7! ways to select the partners for the horses. It follows that the answer is 4 × 5 × 7! = 100800.