Question

十把椅子均匀地围绕一张圆桌摆放，并按顺时针编号为 $1$ 到 $10$ 。五对夫妻要坐在这些椅子上，男女交替，并且没有人坐在自己配偶的旁边或正对面。共有多少种座位安排？

Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10.$ Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?

$240$

$360$

$480$

$540$

$720$

答案：C

知识点：环形排列有限制的排列分类讨论

难度评级：1870

解答：

先安排女性。第一位女性可坐任意 $10$ 把椅子，而男女必须交替，所以其余女性在剩下四个同类座位中有 $4!$ 种排法，共 $10\cdot 4!=240$ 种安排。

固定一位女性在椅子 $1$ 。她的配偶必须坐在椅子 $4$ 或椅子 $8$ ；每个选择都会一致地迫使其他男性的位置。因此每种女性安排恰有 $2$ 种有效男性安排。

总数为 $2\cdot 240=480$ 。

所以正确答案是 C。

Seat the women first. The first woman may take any of the $10$ chairs, and since seats alternate, the remaining women fill their four seats in $4!$ ways, giving $10\cdot 4!=240$ arrangements.

Fix a woman in chair $1.$ Her spouse must sit in chair $4$ or chair $8;$ each choice then forces the placement of every other man consistently. So each seating of the women yields exactly $2$ valid seatings of the men.

The total is $2\cdot 240=480.$

Thus, the correct answer is C.

2008 AMC 10B 第 21 题

答案：C

难度评级：1870

解答：

其他年份的第 21 题