In a group of 19 people each must shake hands with the others how many handshakes

It's 18 + 17 + 16 ... + 1.

This is because each person shakes hands with 18 other people, but then they have already shaken hands with a certain amount of people, and that number goes up by one every single time.

The sum of that is 171.

So 171 handshakes occur.