Handshakes Counting Question [Math]

Hi there. In this math post I cover the question type that involves the amount of handshakes taken when people meet each other.

Handshake Question

You have ten people all meeting each other for the first time. Everyone shakes each others hand once. How many handshakes occur between 10 people?

This question can be super confusing at first. You may not know where to start. Start with the simplest case of 2 people.

Two People Case

When two people meet each other for the first time, there is only one handshake.

Three People Case

You have persons A, B and C. Person A shakes hands with person B, person A with person C and person B with C. This is 3 handshakes. You can view this as 1 + 2 = 3.

Four People Case

With four people, you have A, B, C and D. The handshakes would be

This would be 6 handshakes. The calculation could be written as 1 + 2 + 3.

Let n be the number of people meeting each other for the first time. The number of handshakes would be:

The original question wanted the number of handshakes from 10 people. This would be:

General Formula

Adding a bunch of numbers together takes a long time if you are not using programming. There is a general formula for adding from n down to 1 where there are n people.

Proof

The formula is based on the formula for an arithmetic series. The difference d is one, a1 = 1 and the last term is a_n = n - 1.

If you have 10 people shaking each others hands for the first time, substitute n = 10 in the formula.

Resource/Reference For More Details