Handshakes at a party of ten couples

Question

Ten married couples attend a party. Every person shakes hands exactly once with every other person present, except their own partner. How many handshakes take place? Give two different ways to count.

Accepted Answer

180 handshakes. Method 1: subtract the missing ones. Among 20 people there are 20 choose 2 = 190 possible handshakes. Exactly 10 of them, one per couple, do not happen. So 190 minus 10 = 180. Method 2: per person. Each person shakes hands with 20 minus 1 (themselves) minus 1 (their partner) = 18 others. Twenty people times 18 counts every handshake twice (once from each end), so the total is 20 × 18 / 2 = 180. The general formula. With n couples, the count is (2n)(2n minus 2)/2 = 2n(n minus 1): 2 × 10 × 9 = 180. For 5 couples it is 40, for 100 couples 19,800. A variant to watch for. "Everyone…

Handshakes at a party of ten couples

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