Handshakes at a party

Question

Ten people arrive at a party. Every person shakes hands with every other person exactly once. How many handshakes take place in total?

Accepted Answer

Each of the 10 people shakes hands with the other 9. Naively that suggests 10 	imes 9 = 90 handshakes. But every handshake involves two people, so this count includes each handshake twice, once from each participant's point of view. Divide by 2: \frac{10 	imes 9}{2} = 45. More generally, the number of handshakes among n people is the number of ways to choose an unordered pair: \binom{n}{2} = \frac{n(n-1)}{2}. You can also see it by adding: the first person makes 9 new handshakes, the next makes 8 new ones (the one with person 1 is already counted), then 7, and so on down to 1. The sum 9 + 8…

Handshakes at a party

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