HMMT 二月 2008 · GEN1 赛 · 第 3 题

HMMT February 2008 — GEN1 Round — Problem 3

[ 3 ] There are 5 dogs, 4 cats, and 7 bowls of milk at an animal gathering. Dogs and cats are distin- guishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken? 2 2 4 4 17

解析

[ 3 ] There are 5 dogs, 4 cats, and 7 bowls of milk at an animal gathering. Dogs and cats are distin- guishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken? Answer: 20 Since there are 9 dogs and cats combined and 7 bowls of milk, there can only be one dog-cat pair, and all the other pairs must contain a bowl of milk. There are 4 × 5 ways of selecting the dog-cat pair, and only one way of picking the other pairs, since the bowls of milk are indistinguishable, so the answer is 4 × 5 = 20. 17 2 2 4 4