Sample Problem

Four friends stand in a room. If each person shakes hands with everyone exactly once, how many handshakes occur?



The first person can shake hands with each of the other three people. The second person has already shaken hands with the first person, so he/she can shake hands with two other people. Similarly, the third person only has one other person to shake hands with, and the fourth person has none. This is a total of 3 + 2 + 1 = 6 handshakes.

Another way to think about this is by counting the number of ways of choosing two people who shake hands. The first person can be any of the four, and he/she can choose one of the three others to shake hands with. Since the order in which the people are selected does not matter, we divide by two: 4 × 3 ÷ 2 = 6.

We can apply this method to other problems in which objects (not necessarily people) “shake hands” with each other.