The Handshake Problem

Overview

Counting handshakes is the friendliest introduction to combinations — and to the trap of double-counting.

How to solve The Handshake Problem

1. Each of the 6 people shakes hands with the other 5, suggesting 6 × 5 = 30.
2. But A-shakes-B is the same event as B-shakes-A, so every handshake is counted twice.
3. Divide by 2: 30 ÷ 2 = 15 handshakes.

The key insight

Choosing an unordered pair from n items is n(n−1)/2 — the '÷2' removes the double-counting of order.

Variations & echoes

• The same count gives edges in a complete graph and clinks in a group toast.
• It generalises to the binomial coefficient C(n, 2).

Frequently asked questions