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2017 AMC 10A Problem 8

Problem 8 of 25EasierCombinatorics

At a gathering of 3030 people, there are 2020 people who all know each other and 1010 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?

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Solution

Each of the 1010 people shake hands with each of the 2020 people. This results in 10⋅20=20010 \cdot 20 = 200 handshakes. There are also (102)=45\binom{10}{2} = 45 handshakes within the 1010 people (every pair of people shake hands). Therefore, the total number of handshakes is 200+45=245.200 + 45 = 245. Thus, B is the correct answer.
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Tagged: counting pairs · combinations

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