2017 AMC 10A Problem 8

Question

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

Count handshakes involving the ten people who know no one

Avoid double-counting by thinking in pairs

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.

Problem 8 in other years