Assume there are 10 people in a room, including you. each person in the room must shake hands one time, and only one time, with all the other people in the room. how many handshakes will occur?

You calculate the number of ways that 2 people can be chosen from 10.
The formula:
n! / r! * (n-r)! =
10! / 2! * 8! =
10 * 9 * 8! / 2 * 8! =
10 * 9 / 2 =
45 handshakes

Source:
http://www.1728.org/combinat.htm