If you have six people in a room, and each person shakes hands with every other person
exactly once, how many total handshakes happen?
(a) 6 (b) 12 (c) 15 (d) 21 (e) 720

The way you can show this is;
(people labeled 1-6)
6 + 5 | 5 + 4 | 4 + 3 | 3 + 2 | 2 + 1
6 + 4 | 5 + 3 | 4 + 2 | 3 + 1
6 + 3 | 5 + 2 | 4 + 1
6 + 2 | 5 + 1
6 + 1 | 

There are 30 possible combinations, but if they shake hands with every other person, the number is halved.

This means that there are 15 handshakes total.