2017 AMC 10A Problem 8

Below is the professionally curated solution for Problem 8 of the 2017 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 10A solutions, or check the answer key.

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Concepts:counting pairs combinations

Difficulty rating: 1020

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 within the group?

$240$

$245$

$290$

$480$

$490$

Solution:

Each of the $10$ people shake hands with each of the $20$ people. This results in $10 \cdot 20 = 200$ handshakes.

There are also $\binom{10}{2} = 45$ handshakes within the $10$ people (every pair of people shake hands).

Therefore, the total number of handshakes is $200 + 45 = 245.$

Thus, B is the correct answer.

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