Question

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

Answer 1

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 `(AB)`.

If we have 3 people, there are 3 handshakes `(AB, AC, BC)`.

If we have 4 people, there are 6 handshakes `(AB, AC, AD, BC, BD, CD)`.

If we have 5 people, there are 10 handshakes `(AB, AC, AD, AE, BC, BD, BE, CD, CE, DE)`.

See that we can express the number of handshakes as the sum of consecutive positive integers, starting with 1, i.e. `1+2+3+...+(n-1)` and the number of people present is `n`

Let's test this with 5 people. We have `1+2+3+4=10` handshakes. `n-1=4=>n=5` which is the number of people.

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

`1+2+3+4+5+6+7+8+9+10+11=66=>`

`=>n-1=11=>n=12`