2017 AMC 10A Problems/Problem 8

Revision as of 16:41, 8 February 2017 by Always correct (talk | contribs) (Solution)

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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