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 $10\cdot20$ = 200. From there, we calculate how many handshakes occurred between the people who don’t know each other. This is simply counting how many ways to choose two people to shake hands, or $\binom{10}{2} = 45$ . Thus the answer is $200 + 45 = \boxed{\textbf{(B)} 245}$ .

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2017 AMC 10A Problems/Problem 8

Problem

Solution