2017 AMC 10 A
Complete problem set with solutions and individual problem pages
At a gathering of 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? (2017 AMC 10A Problem, Question#8)
- A.
- B.
- C.
- D.
- E.
Each one of the ten people has to shake hands with all the other people they don't know. So . 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 . Thus the answer is .
We can also use complementary counting. First of all, handshakes or hugs occur. Then, if we can find the number of hugs, then we can subtract it from to find the handshakes. Hugs only happen between the people who know each other, so there are hugs. .
We can focus on how many handshakes the people get.
The person gets handshakes.
gets .
And the receives handshakes.
We can write this as the sum of an arithmetic sequence.
. Therefore, the answer is .
