Question

六把椅子围着一张圆桌摆放。两名学生和两名老师随机选择其中四把椅子坐下。两名学生坐在相邻两把椅子上，并且两名老师也坐在相邻两把椅子上的概率是多少？

Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{2}{9}$

$\dfrac{3}{13}$

$\dfrac{1}{4}$

答案：B

知识点：环形排列基本概率

难度评级：1350

小提示：

数一数把一对椅子给学生、另一对椅子给老师的方法数

Count the ways to give the students one pair of chairs and the teachers another pair

大提示：

共有 $6$ 对相邻椅子；固定学生的一对后，数留给老师的相邻椅子对

There are $6$ adjacent pairs; after fixing the students’ pair, count the adjacent pairs left for the teachers

解答：

给学生选择 $2$ 把椅子、给老师选择 $2$ 把椅子，共有 $\binom{6}{2}\binom{4}{2} = 15 \cdot 6 = 90$ 种等可能结果。

圆桌周围有 $6$ 对相邻椅子。学生可以得到任意一对相邻椅子；在剩下的 $4$ 把椅子中，老师正好有 $3$ 对相邻椅子可选。因此有 $6 \cdot 3 = 18$ 种有利结果。

所求概率为 $\dfrac{18}{90} = \dfrac{1}{5}$ 。

因此，正确答案是 B。

Choosing $2$ chairs for the students and $2$ for the teachers gives $\binom{6}{2}\binom{4}{2} = 15 \cdot 6 = 90$ equally likely outcomes.

A round table has $6$ adjacent pairs of chairs. Give the students any adjacent pair; among the remaining $4$ chairs there are exactly $3$ adjacent pairs for the teachers. That is $6 \cdot 3 = 18$ favorable outcomes.

The probability is $\dfrac{18}{90} = \dfrac{1}{5}.$

Thus, the correct answer is B.

2025 AMC 12A 第 6 题

答案：B

难度评级：1350

小提示：

大提示：

解答：

其他年份的第 6 题