1989 AMC 12 第 24 题

Question

五个人围坐在一张圆桌旁。设

f\ge0

为至少与一名女性相邻而坐的人数，

m\ge0

为至少与一名男性相邻而坐的人数。可能的有序数对

(f,m)

一共有多少个？

Five people are sitting at a round table. Let

f\ge0

be the number of people sitting next to at least one female and

m\ge0

be the number of people sitting next to at least one male. The number of possible ordered pairs

(f,m)

is

7

8

9

10

11

按照女性人数对座位安排进行分类

Classify arrangements by the number of females

有两名女性时，分别考虑她们相邻和不相邻的情形；再利用对称性得到有三名女性的情形

With two females, separate the adjacent and nonadjacent cases; obtain three-female cases by symmetry

当女性人数分别为

0,1,2

时，可能的数对依次为

\begin{aligned} &(0,5),\ (2,5),\\ &(4,5),\ (3,4) \end{aligned}\text{，}

其中有两名女性的两种情形分别对应她们相邻和不相邻。交换两种性别，可得

\begin{aligned} &(5,0),\ (5,2),\\ &(5,4),\ (4,3) \end{aligned}\text{。}

这些是

8

个不同的有序数对。

所以正确答案是 B。

For

0,1,2

females, the possible pairs are respectively

\begin{aligned} &(0,5),\ (2,5),\\ &(4,5),\ (3,4), \end{aligned}

where the two-female cases distinguish adjacent from nonadjacent females. Swapping the sexes gives

\begin{aligned} &(5,0),\ (5,2),\\ &(5,4),\ (4,3). \end{aligned}

These are

8

distinct ordered pairs.

Thus the correct answer is B.

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