Overview
Counting the number of ways to arrange or choose items — the key is whether order matters.
The two tools
- Permutations (ⁿPᵣ) — order matters (arrangements):
ⁿPᵣ = n! ÷ (n − r)!
- Combinations (ⁿCᵣ) — order does not matter (selections):
ⁿCᵣ = n! ÷ [r!(n − r)!]
Factorial: n! = n × (n − 1) × … × 1, and 0! = 1.
Which to use?
- "A team / a group / a committee" → combination (order doesn't matter).
- "A code / a line-up / first, second, third" → permutation (order matters).
Worked example
Choose 3 students from 10 for a committee: ¹⁰C₃ = 10! ÷ (3! × 7!) = 120. If instead they take 1st, 2nd, 3rd place: ¹⁰P₃ = 720.
Common mistakes
- Using permutations for an unordered selection (or vice versa).
Exam tips
- ⁿCᵣ is always smaller than ⁿPᵣ (you've removed the orderings).