HMMT 二月 2007 · GEN1 赛 · 第 9 题

英文解析

9. [ 6 ] I have four distinct rings that I want to wear on my right hand hand (five distinct fingers.) One ofthese rings is a Canadian ring that must be worn on a finger by itself, the rest I can arrange however
   I want. If I have two or more rings on the same finger, then I consider different orders of rings alongthe same finger to be different arrangements. How many different ways can I wear the rings on myfingers?
   Answer: 600 . First we pick the finger for the Canadian ring. This gives a multiplicative factor of 5.
   For distributing the remaining 3 rings among 4 fingers, they can either be all on the same finger (4 · 3!
   ( ) ( )
   4 3
   ways), all on different fingers ( · 3! ways), or two on one finger and one on another (4 · · 2! · 3
   3 2
   ways.) Therefore, I have 5 · (24 + 24 + 72) = 600 choices.
   4 3