Patrick Honner in Quanta:
Say you’re at a party with nine other people and everyone shakes everyone else’s hand exactly once. How many handshakes take place?
This is the “handshake problem,” and it’s one of my favorites. As a math teacher, I love it because there are so many different ways you can arrive at the solution, and the diversity and interconnectedness of those strategies beautifully illustrate the power of creative thinking in math.
One solution goes like this: Start with each person shaking every other person’s hand. Ten people, with nine handshakes each, produce 9 × 10 = 90 total handshakes. But this counts every handshake twice — once from each shaker’s perspective — so the actual number of handshakes is 902=45. A simple and lovely counting argument for the win!
There’s also a completely different way to solve the problem.
More here.