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：1600

解答：

先让第一名学生坐任意位置。第二名学生坐到相邻椅子的概率为 $\tfrac{2}{5}$ ，因为剩下五把椅子中有两把与第一名学生相邻。接着老师占据剩下四把椅子中的两把。在 $\binom{4}{2} = 6$ 种选法中，恰好有 $3$ 对是相邻椅子。因此所求概率为 $\tfrac{2}{5} \cdot \tfrac{3}{6} = \tfrac{1}{5}$ 。所以正确答案是 B。

Seat the first student anywhere. The second student lands next to them with probability $\tfrac{2}{5},$ since two of the remaining five chairs are adjacent. Now the teachers fill two of the four leftover chairs. Of the $\binom{4}{2} = 6$ ways to do that, exactly $3$ are adjacent pairs. So the probability is $\tfrac{2}{5} \cdot \tfrac{3}{6} = \tfrac{1}{5}.$ Therefore, the answer is B.

2025 AMC 10A 第 14 题

答案：B

难度评级：1600

解答：

其他年份的第 14 题