Question

一排九把椅子将由六名学生和阿尔法、贝塔、伽马三位教授就座。三位教授比六名学生先到，并决定选择座位，使每位教授的两边都是学生。阿尔法、贝塔和伽马三位教授有多少种选座方式？

Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs?

$12$

$36$

$60$

$84$

$630$

答案：C

知识点：seating arrangements nonconsecutive selections

难度评级：1900

小提示：

教授的座位不能在两端，也不能彼此相邻

Professor chairs cannot be endpoints or adjacent to one another

大提示：

先从第 $2$ 把至第 $8$ 把椅子中选出三个互不相邻的位置，再安排三位教授

First choose three nonconsecutive positions from chairs $2$ through $8$ , then assign the professors

解答：

教授必须占据第 $2,3,\ldots,8$ 把椅子中的三个互不相邻的位置。这种 $3$ 元子集从连续 $7$ 个位置中选取，其数量为 $\binom{7-3+1}{3}=\binom53=10\text{。}$ 三位不同的教授可以用 $3!=6$ 种方式安排到所选椅子上，共有 $10\cdot6=60$ 种。因此正确答案是 C。

The professors must occupy three nonconsecutive positions among chairs $2,3,\ldots,8.$ The number of such $3$ -subsets of $7$ consecutive positions is $\binom{7-3+1}{3}=\binom53=10.$ The three distinct professors can be assigned to the selected chairs in $3!=6$ ways, for $10\cdot6=60.$ Thus the correct answer is C.

1994 AMC 12 第 22 题

答案：C

难度评级：1900

小提示：

大提示：

解答：

其他年份的第 22 题