Permutations vs combinations: a simple way to tell them apart
Does order matter? A clear guide to choosing between permutations and combinations, with card examples, formulas and common mistakes.
· 5 min read
The one-question test
Ask: if I rearrange the chosen items, do I get a different outcome? If yes, order matters and you count permutations. If no, you count combinations.
Permutations: order matters
Dealing five cards to five labelled seats, where each seat getting a different card counts as different, is a permutation: P(52, 5) = 52 × 51 × 50 × 49 × 48 = 311,875,200.
P(n, k) = n! / (n − k)!Combinations: order does not matter
The set of five cards in a hand is a combination: C(52, 5) = 2,598,960. Each set appears in 5! = 120 different orders, which is why combinations equal permutations divided by k!.
C(n, k) = n! / (k! (n − k)!)Mistakes to avoid
- Using n × n instead of n × (n − 1): without replacement the choices shrink.
- Forgetting to divide by k! when order should not matter.
- Using floating-point numbers for huge factorials. This site uses exact big integers.