2017 AMC 12A 第 5 题

Question

在一个

30

人聚会上，有

20

个人彼此都认识，还有

10

个人谁也不认识。彼此认识的人拥抱，彼此不认识的人握手。一共会发生多少次握手？

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

那

20

个彼此认识的人只会和

10

个陌生人握手

The

20

acquaintances only shake hands with the

10

strangers

把每个人的握手次数相加，再除以

2

。

Add up each person’s handshakes and divide by

2

那

20

个彼此认识的人只和

10

个陌生人握手。那

10

个陌生人中的每个人都和其他

29

人握手。

把握手次数相加并除以

2

（每次握手涉及两个人），得到

\begin{aligned} &\dfrac{1}{2}(20\cdot10+10\cdot29) \\ &=\dfrac{1}{2}(200+290)=245 \end{aligned}\text{。}

所以正确答案是 B。

Each of the

20

people who know each other shakes hands with only the

10

strangers. Each of the

10

strangers shakes hands with all

29

other people.

Summing handshake counts and dividing by

2

(each handshake involves two people) gives

\begin{aligned} &\dfrac{1}{2}(20\cdot10+10\cdot29) \\ &=\dfrac{1}{2}(200+290)=245. \end{aligned}

Thus, the correct answer is B.

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