1990 AMC 12 第 16 题

Question

在乔治·华盛顿的一次聚会上，每位男士都与除自己配偶以外的所有人握手，而女士之间互不握手。如果有

13

对夫妇参加，那么这

26

人之间一共握了多少次手？

At one of George Washington’s parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If

13

married couples attended, how many handshakes were there among these

26

people?

78

185

234

312

325

先考虑

26

位宾客中所有无序的两人组合

Begin with all unordered pairs of the

26

guests

去掉夫妻组合和由两位女士组成的组合

Remove spouse pairs and pairs consisting of two women

所有可能的两人组合共有

\binom{26}{2}=325

个。去掉

13

对夫妻和

\binom{13}{2}=78

个由两位女士组成的组合。握手次数为

325-13-78=234\text{。}

所以正确答案是 C。

There are

\binom{26}{2}=325

possible pairs. Exclude the

13

married pairs and the

\binom{13}{2}=78

pairs of women. The number of handshakes is

325-13-78=234.

Thus the correct answer is C.

其他年份的第 16 题