HMMT 二月 2000 · ADV 赛 · 第 6 题
HMMT February 2000 — ADV Round — Problem 6
题目详情
- 6 p eople ea h ha v e a hat. If they sh u e their hats and redistribute them, what is the probabilit y that exa tly one p erson gets their o wn hat ba k? p p a b 3 2 2 2 2
解析
- There are 6 p eople that ould get their hat ba k, so w e m ust m ultiply 6 b y the n um b er of w a ys that the other 5 p eople an arrange their hats su h that no one gets his/her hat ba k. So, the n um b er of w a ys this will happ en is (6 derangemen t of 5), or 6 44 = 264. Sin e there are 6! = 720 p ossible arrangemen ts of hats, the probabilit y of exa tly one 11 264 p erson getting their hat ba k is = . 720 30