AMC10 2017 A

AMC10 2017 A · Q8

AMC10 2017 A · Q8. It mainly tests Basic counting (rules of product/sum).

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?

在30人的聚会上，有20人互相都认识，还有10人谁都不认识。认识的人拥抱，不认识的人握手。发生了多少次握手？

(A) 240 240

(B) 245 245

(D) 480 480

(E) 490 490

Answer

Correct choice: (B)

正确答案：（B）

Solution

Each of the 20 people who know each other shakes hands with 10 people. Each of the 10 people who know no one shakes hands with 29 people. Because each handshake involves two people, the number of handshakes is\n\[\frac{1}{2}(20 \cdot 10 + 10 \cdot 29) = 245.\]

20个互相认识的人中的每个人与10人握手。10个谁都不认识的人中的每个人与29人握手。因为每次握手涉及两个人，握手次数为\n\[\frac{1}{2}(20 \cdot 10 + 10 \cdot 29) = 245.\]

Topics

CP.001 Basic counting (rules of product/sum)