Permutation Calculator – P(n,r) and Combination C(n,r)
Calculate permutations P(n,r) and combinations C(n,r) with step-by-step solutions.
Permutations vs. Combinations
A permutation is an arrangement of items where order matters. A combination is a selection where order does not matter. Choosing 3 items from 5: P(5,3) = 60 ordered arrangements, C(5,3) = 10 unordered groups.
| n (items) | r (selected) | P(n,r) ordered | C(n,r) unordered |
|---|---|---|---|
| 5 | 2 | 20 | 10 |
| 5 | 3 | 60 | 10 |
| 10 | 3 | 720 | 120 |
| 52 | 5 | 311,875,200 | 2,598,960 |
Formulas: P(n,r) = n! / (n−r)! and C(n,r) = n! / (r! × (n−r)!). Real-world examples: a 4-digit PIN from 10 digits has P(10,4) = 5,040 possible ordered codes. A committee of 3 from 10 candidates has C(10,3) = 120 ways — the order of selection doesn't change who serves.
When do I use permutation vs combination?
Use permutation when order matters: passwords, race finishes, seating arrangements. Use combination when order doesn't matter: lottery numbers, committee selection, card hands.
What is 0! (zero factorial)?
By mathematical convention, 0! = 1. This makes formulas like C(n,0) = 1 consistent — there is exactly one way to choose nothing from a set.