2017 AMC 10A Problems/Problem 8
At a gathering of people, there are people who all know each other and 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?
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 who don't know anybody get.
The first person gets handshakes with other people not him/herself, the second person gets handshakes with other people not him/herself and not the first person, ..., and the tenth receives handshakes with other people not him/herself and not the first, second, ..., ninth person. We can write this as the sum of an arithmetic sequence:
Therefore, the answer is
First, we can find out the number of handshakes that the people who don't know anybody share with the other people. This is simply . Next, we need to find out the number of handshakes that are shared within the people who don't know anybody. Here, we can use the formula , where is the number of people being counted. The reason we divide by is because counts the case where the person shakes hands with the person the case where the shakes hands with the (and these 2 cases are the same). Thus, plugging gives us . Adding up the 2 cases gives us
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