How many handshakes?
At a party everyone shakes hands with everyone else once. Change the number of people from 2 to 12 and count the handshakes, then find the rule that works for any group.
Core concept
In a group of n people each person shakes n − 1 hands, but every handshake is shared by two people, so there are n(n − 1) ÷ 2 handshakes: the number grows much faster than the group
Topic: Maths in everyday life
Five friends meet and each shakes hands with every other friend once. How many handshakes are there? Each of the 5 people shakes 4 hands, which gives 5 × 4 = 20. But that counts every handshake twice, once for each of the two hands in it. So there are 20 ÷ 2 = 10 handshakes. The same thinking works for any number n of people: n(n − 1) ÷ 2. The same rule counts the matches in a league where every team plays every other team once, and the lines joining every pair of points.
What you will learn
- Count handshakes in a small group systematically.
- Explain why we divide by 2.
- Use the rule n(n − 1) ÷ 2 for any group.
Before you start
- Multiply and divide whole numbers.
Materials
- A few friends, or dots on paper joined by lines
Equipment
- None
Procedure
- Record the handshakes for 5 people.
- Record the handshakes for 2 people.
- Record the handshakes for a class of 10 people.
How this simulation is simplified
- Every pair of people shakes hands exactly once.