Permutations & Combinations
Calculate permutations P(n,r) and combinations C(n,r) — the number of ways to arrange or select r items from a set of n. Also computes factorials.
Permutations & Combinations Formulas
P(n,r) = n! / (n − r)! — ordered arrangements
C(n,r) = n! / (r! × (n − r)!) — unordered selections
C(n,r) = P(n,r) / r! — combinations are always ≤ permutations
When to use which
| Permutations | Passwords, PIN codes, race rankings, seating arrangements |
| Combinations | Lottery tickets, team selection, card hands, committee formation |
How to use
Permutations P(n,r)
The number of ways to arrange r items chosen from n, where the order matters. Example: how many 3-letter codes can be made from 10 letters? P(10,3) = 720.
Combinations C(n,r)
The number of ways to choose r items from n, where order does NOT matter. Example: how many 5-card hands from a 52-card deck? C(52,5) = 2,598,960.
When to use which
Use permutations for ordered arrangements (passwords, rankings, race positions). Use combinations for unordered selections (lottery, team selection, card hands).
Formula
P(n,r) = n! / (n−r)!
C(n,r) = n! / (r! × (n−r)!)
C(n,r) = P(n,r) / r! — combinations are always smaller
Tips
- •C(52,5) = 2,598,960 poker hands. Of these, only 4 are royal flushes — odds of ~1 in 650,000.
- •C(n,r) = C(n, n−r) — choosing 3 from 10 equals choosing 7 from 10.
- •Pascal's triangle rows give the combination values C(n,0), C(n,1), … C(n,n).