2017 AMC 12A Problem 5

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

The

20

acquaintances only shake hands with the

10

strangers

Add up each person’s handshakes and divide by

2

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.

Problem 5 in other years