2017 AMC 12A Problem 5

Below is the professionally curated solution for Problem 5 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

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Concepts:graph theory basic counting

Difficulty rating: 1270

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?

$240$

$245$

$290$

$480$

$490$

Solution:

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 $\dfrac{1}{2}(20\cdot10+10\cdot29)=\dfrac{1}{2}(200+290)=245.$

Thus, the correct answer is B.

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