At a party, every person shakes hands with every other person exactly once.
Question: If there are n people at the party, how many handshakes happen in total?
Start small — work it out for 2, 3, 4 people — then spot the pattern and find a general formula.
Each person shakes hands with (n − 1) others.
Total count = n × (n − 1) — but this double-counts each handshake (A shaking B's hand = B shaking A's hand).
So: Total handshakes = n(n − 1) / 2
This is also written as C(n, 2) — "n choose 2" — the number of ways to pick 2 people from n.
People (n)
Handshakes
Formula check
2
1
2×1/2 = 1 ✓
3
3
3×2/2 = 3 ✓
4
6
4×3/2 = 6 ✓
5
10
5×4/2 = 10 ✓
10
45
10×9/2 = 45 ✓
📝 Questions
1
List all handshakes when there are 4 people: A, B, C, D. Write each pair (e.g. A–B). How many are there? 2 marks
2
Fill in the table: number of handshakes for n = 2, 3, 4, 5, 6. Spot the pattern — what type of numbers are these? 3 marks
n (people)
2
3
4
5
6
Handshakes
3
With n people, each person shakes hands with (n − 1) others. The total n(n − 1) double-counts each handshake. Explain why it double-counts, and write the corrected formula. 3 marks
4
Use your formula to find the number of handshakes for: (a) n = 12, (b) n = 20, (c) n = 100. 3 marks
5
At a party there were exactly 55 handshakes. How many people were at the party? Show your working. 3 marks
6
Extension: The formula n(n−1)/2 is also written as C(n,2). In a football league with 20 teams, each team plays every other team twice (home and away). How many matches are played in total? 6 marks
Answer Key
🤝 Handshake Problem
1.A–B, A–C, A–D, B–C, B–D, C–D → 6 handshakes.
2.1, 3, 6, 10, 15 — these are triangular numbers.
3.When A shakes B's hand, we count it once for A and once for B — each handshake appears twice. Divide by 2: n(n−1)/2.