HMMT 十一月 2011 · 冲刺赛 · 第 13 题

HMMT November 2011 — Guts Round — Problem 13

[ 8 ] Tac is dressing his cat to go outside. He has four indistinguishable socks, four indistinguishable shoes, and 4 indistinguishable show-shoes. In a hurry, Tac randomly pulls pieces of clothing out of a door and tries to put them on a random one of his cat’s legs; however, Tac never tries to put more than one of each type of clothing on each leg of his cat. What is the probability that, after Tac is done, the snow-shoe on each of his cat’s legs is on top of the shoe, which is on top of the sock?

解析

[ 8 ] Tac is dressing his cat to go outside. He has four indistinguishable socks, four indistinguishable shoes, and 4 indistinguishable show-shoes. In a hurry, Tac randomly pulls pieces of clothing out of a door and tries to put them on a random one of his cat’s legs; however, Tac never tries to put more than one of each type of clothing on each leg of his cat. What is the probability that, after Tac is done, the snow-shoe on each of his cat’s legs is on top of the shoe, which is on top of the sock? 1 Answer: On each leg, Tac’s cat will get a shoe, a sock, and a snow-shoe in a random order. 1296 1 1 Thus, the probability that they will be put on in order for any given leg is = . Thus, the probability 3! 6 ( ) 4 1 1 that this will occur for all 4 legs is = . 6 1296