Study guides › Level 6 · Master

Handshakes (counting pairs)

Level 6 · Master · for questions like “12 friends each shake hands with every other friend once. How many handshakes?”

💡 The idea

Each person shakes hands with everyone else (n − 1 people). But every handshake has two people in it, so it gets counted twice.

handshakes = n × (n − 1) ÷ 2

🪜 The steps

1. Multiply the number of people by one less.
2. Halve it.

✏️ Worked examples

1. 12 friends: 12 × 11 = 132, then ÷ 2 = 66.

2. 5 friends: 5 × 4 ÷ 2 = 10. You can check by listing them.

⚠️ Watch out

• Don't forget to halve. Ann shaking Ben's hand is the same handshake as Ben shaking Ann's.
• The same pattern works for games where everyone plays everyone once, or lines joining dots.

🎯 Try these

1. 6 friends
2. 20 friends
3. 10 teams, each playing every other team once

Answers: 1) 15 2) 190 3) 45 games

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.