# Count the handshakes: visual puzzle and solution | SwaVid

Ten people each shake hands with every other person once. How many handshakes happen? See the complete step-by-step solution.

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# Count the handshakes

## 45 handshakes.

### Count from one person

### Remove the double count

### Check by pairs

SwaVid visual puzzle

Ten people each shake hands with every other person once. How many handshakes happen?

Detailed solution

Each handshake involves two people, so counting from every person's point of view counts each handshake twice.

Each person shakes hands with the other 9.

A handshake between A and B was counted once for A and once for B.

This is the number of ways to choose 2 people from 10.

Answering 90 forgets that a handshake is shared by two people; answering 100 lets people shake their own hands.

- 1 Count from one person Each person shakes hands with the other 9. 10 × 9 = 90 person-handshakes
- 2 Remove the double count A handshake between A and B was counted once for A and once for B. 90 ÷ 2 = 45
- 3 Check by pairs This is the number of ways to choose 2 people from 10. 10 × 9 ÷ 2 = 45

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