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栖息的鸟

Sitting Birds

专题
Probability / 概率
难度
L2

题目详情

两只棕色鸟和六只红色鸟在空中飞，随后随机排成一行落在电话线上。两只棕色鸟恰好挨在一起的概率是多少？

Two brown and six red birds are flying around the sky and suddenly all land randomly in line on a telephone wire. What is the probability that the two brown birds are side by side?

解析

把两只棕色鸟看作一个整体“块”：

  • 8 只不同鸟的总排列数为 8!8!
  • 若要求两只棕鸟相邻，则把这对棕鸟视为一个整体，此时有 7 个对象可排，因此有 7!7! 种排法；而块内两只棕鸟还可以互换顺序，所以再乘 2!2!

因此有利情况数为 2!×7!2! \times 7!，所以

Pr(brown birds adjacent)=2!7!8!=28=14\text{Pr(brown birds adjacent)} = \frac{2!7!}{8!} = \frac{2}{8} = \boxed{\frac{1}{4}}

Original Explanation

Treat the two brown birds as a single "block":

  • Total arrangements of 8 distinct birds: 8!8!
  • Arrangements with the brown birds adjacent: view the pair as one block => 7! ways to place the 7 items, and 2! ways to order the two browns within the block.

So favorable =2!×7!= 2! \times 7!, and

Pr(brown birds adjacent)=2!7!8!=28=14\text{Pr(brown birds adjacent)} = \frac{2!7!}{8!} = \frac{2}{8} = \boxed{\frac{1}{4}}