# At a party 66 handshakes occurred. Each person shook hands exactly once with each of the other people present. How many people were present?

##### 1 Answer

12

#### Explanation:

Let's start with small numbers of people and handshakes and move from there. I'll represent people with letters to show the handshakes:

If we have 2 people, there is 1 handshake

If we have 3 people, there are 3 handshakes

If we have 4 people, there are 6 handshakes

If we have 5 people, there are 10 handshakes

See that we can express the number of handshakes as the sum of consecutive positive integers, starting with 1, i.e.

Let's test this with 5 people. We have

So what we need to do is add up to 66 and we'll be able to find the number of people: