Question

$10$ 个人等间距站在一个圆周上。每个人恰好认识另外 $9$ 人中的 $3$ 人：站在自己两旁的 $2$ 人，以及圆周正对面的那个人。有多少种方法把这 $10$ 人分成 $5$ 对，使每一对中的两人互相认识？

There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other?

$11$

$12$

$13$

$14$

$15$

答案：C

知识点：图论分类讨论

难度评级：1820

解答：

将圆周上的人编号，并按正对面配对的数量分类计数。

若没有正对面配对，则所有人都必须与圆周相邻者配对，恰有 $2$ 种交替相邻配对。

若有一对正对面配对，可用 $5$ 种方式选择这对。剩下的人形成两条各含四个顶点的路径，每条路径只有一种相邻完美匹配，所以给出 $5$ 种。

若有两对或四对正对面配对，剩余相邻配对路径中会出现奇数长度部分，因此不可能完美匹配。

若有三对正对面配对，没选的两对正对面位置必须在五个正对面位置中相邻；否则剩下的人不能用相邻配对匹配。有 $5$ 种相邻选择，所以有 $5$ 种匹配。

若五对全是正对面配对，则有 $1$ 种匹配。总数为 $2+5+5+1=13.$

所以正确答案是 C。

Label the people around the circle. Count by the number of pairs of opposite people.

With no opposite pairs, everyone must be paired with a neighbor around the 10-cycle. There are exactly $2$ alternating neighbor matchings.

With one opposite pair, choose that pair in $5$ ways. The remaining people form two paths of four vertices, and each path has only one perfect matching by neighbor pairs, so this gives $5$ matchings.

With two or four opposite pairs, the remaining neighbor-pairing paths have odd length somewhere, so no perfect matching is possible.

With three opposite pairs, the two opposite pairs not chosen must be adjacent around the five opposite-pair positions; otherwise the remaining people cannot be matched by neighbor pairs. There are $5$ adjacent choices for the two unchosen opposite pairs, so there are $5$ matchings.

With all five opposite pairs, there is $1$ matching. The total is $2+5+5+1=13.$

Thus, the correct answer is C .

2020 AMC 10B 第 17 题

答案：C

难度评级：1820

解答：

其他年份的第 17 题