2017 AMC 10A Problems/Problem 8

Problem

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 a hug, and people who do not know each other shake hands. How many handshakes occur?

$\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

Solution

Each one of the ten people has to shake hands with all the 20 other people they don’t know. So $10\times 20$ = 200. From there you also have to calculate how many handshakes occurred between the people who don’t know each other. Each person out of the 10 has to shake hands with 9 other people. That’s 90, however you have to take into account the overlap. For example, there's 10 people so...

Person 1 shakes hands with people- 2,3,4,5,6,7,8,9,10 And person 2 shakes hands with people- 1,3,4,5,6,7,8,9,10 and so on.

There's overlap. Person 1 shakes hands with person 2 and then person 2 shakes hands with person 1. So 90 needs to be divided by 2 so the overlap is taken into account. From there, add 200 + 45 to get the answer. (B) 245.

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2017 AMC 10A Problems/Problem 8

Problem

Solution