2017 AMC 10A 第 8 题

Question

在一个

30

人的聚会上，有

20

个人彼此都认识，另有

10

个人谁也不认识。认识的人拥抱，不认识的人握手。这个群体中会发生多少次握手？

At a gathering of

30

people, there are

20

people who all know each other and

10

people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?

240

245

290

480

490

数与那十个谁也不认识的人有关的握手。

Count handshakes involving the ten people who know no one

用成对计数避免重复计算。

Avoid double-counting by thinking in pairs

那

10

个谁也不认识的人，每人都与

20

个彼此认识的人握手，共

10 \cdot 20 = 200

次。

这

10

个人彼此之间也都不认识，所以他们之间还有

\binom{10}{2} = 45

次握手。

总数为

200 + 45 = 245

。

所以正确答案是 B。

Each of the

10

people shake hands with each of the

20

people. This results in

10 \cdot 20 = 200

handshakes.

There are also

\binom{10}{2} = 45

handshakes within the

10

people (every pair of people shake hands).

Therefore, the total number of handshakes is

200 + 45 = 245.

Thus, B is the correct answer.

其他年份的第 8 题