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?

$240$

$245$

$290$

$480$

$490$

答案：B

知识点：图论基本计数

难度评级：1270

解答：

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

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.

2017 AMC 12A 第 5 题

答案：B

难度评级：1270

解答：

其他年份的第 5 题