圆桌生日顺序

Circular Birthdays

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

7 个人随机坐到圆桌的 7 个座位上（均匀随机），且 7 人年龄互不相同。求他们“按年龄顺序”围坐的概率。

说明：年龄可以按顺时针递增，也可以按逆时针递增。

7 people sit around a circular table uniformly at random. All of them have a distinct age. Find the probability that they sit down at the table in age order. Note that the ages can be increasing in either the clockwise or counter-clockwise directions.

解析

在圆桌上固定最年轻者的位置（旋转等价），剩下 6 人有 $6!=720$ 种排列。

其中只有 2 种满足按年龄顺序（顺时针递增或逆时针递增）。

因此概率为

\frac{2}{6!}=\frac{1}{360}.

Original Explanation

There are $(7-1)! = 6! = 720$ distinct ways for the people to sit at the table with no restrictions. Only two of these seating arrangements have the people in age order: Namely, they are just when they increase CCW or CW. Therefore, our probability is

\dfrac{2}{720} = \dfrac{1}{360}