Five couples were at a party. If each person shook hands exactly once with everyone else except his/her spouse, how many handshakes were exchanged? (Note: One obviously doesn't shake hands with oneself.)
I need to count the total number of handshakes among 10 people, excluding the ones between spouses. I notice that each person shakes hands with 8 other people, since they don't shake hands with themselves or their spouses. That means each person is involved in 8 handshakes. But if I multiply 10 by 8, I get 80, which is too many. That's because I'm counting each handshake twice, once for each person involved. To avoid double-counting, I need to divide by 2. So the total number of handshakes is 10 times 8 divided by 2, which is 40. 