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?

$240$

$245$

$290$

$480$

$490$

答案：B

知识点：数对计数组合

难度评级：1020

解答：

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.

2017 AMC 10A 第 8 题

答案：B

难度评级：1020

解答：

其他年份的第 8 题