How it works
Combinations and permutations both count the ways to choose items. One question decides which you need: does order matter? Picking 3 people for a committee — order doesn’t matter, so it is a combination. Picking a president, vice-president and treasurer — order matters, so it is a permutation. The calculator returns all four cases as exact whole numbers, even when the answer runs to dozens of digits.
combinations: C(n, r) = n! ÷ (r! (n − r)!) permutations: P(n, r) = n! ÷ (n − r)! with repetition: C(n + r − 1, r) · nʳWorked examples. Choosing 3 from 10 people: combinations (a committee) = 10! ÷ (3! × 7!) = 120; permutations (three ranked roles) = 10 × 9 × 8 = 720. With repetition: 220 combinations and 10³ = 1,000 permutations (a 3-digit code). A 6-from-49 lottery has 13,983,816 combinations, and a 5-card poker hand from 52 cards has 2,598,960.
Which one do I need?
| Type | Order matters? | Repeats allowed? | Example |
|---|---|---|---|
| Combination nCr | No | No | Choosing a 5-player team |
| Permutation nPr | Yes | No | Gold, silver and bronze in a race |
| Combination with repetition | No | Yes | Buying 4 donuts from 6 flavours |
| Permutation with repetition | Yes | Yes | A 4-digit PIN: 10,000 options |
Factorials
n! is the product of every whole number from 1 to n: 5! = 120. By convention 0! = 1. Factorials explode quickly — 20! is about 2.4 quintillion — which is why ordinary calculators overflow while this one keeps every digit.
Useful identities
- C(n, r) = C(n, n − r): choosing 3 of 10 is the same as leaving out 7.
- C(n, 0) = C(n, n) = 1 and C(n, 1) = n.
- P(n, r) = C(n, r) × r!, because each group can be ordered r! ways.
- Without repetition, if r > n the answer is 0.
From counting to probability
Probability = favourable outcomes ÷ total outcomes. The chance of a flush (five cards of one suit) is 4 × C(13, 5) ÷ C(52, 5) = 5,148 ÷ 2,598,960 ≈ 0.198%. The chance of winning a 6/49 lottery with one ticket is 1 in 13,983,816.
Convert probabilities to percentages with the percentage calculator, work with fractions in the fraction calculator, and for normal-distribution probabilities use the z-score calculator.