Quick start
- Enter n, the total number of items.
- Enter r, how many you choose.
- Read nPr, nCr and the versions with repetition.
Does order matter?
That one question decides which formula you need. Picking a president, secretary and treasurer from a club is a permutation, because each role is different. Picking a three-person committee is a combination, because the group is the same whichever order you name them in.
The formulas
| Count | Formula | n = 10, r = 3 |
|---|---|---|
| Permutations, nPr | n! ÷ (n − r)! | 720 |
| Combinations, nCr | n! ÷ (r!(n − r)!) | 120 |
| Ordered, with repetition | nʳ | 1,000 |
| Unordered, with repetition | (n + r − 1)! ÷ (r!(n − 1)!) | 220 |
Examples
- A 4-digit PIN with digits 0–9 allows repeats and order matters: 10⁴ = 10,000 codes.
- Choosing 6 numbers from 49 for a lottery: 49C6 = 13,983,816 tickets.
- Gold, silver and bronze among 8 runners: 8P3 = 336 outcomes.
Common questions
What is a factorial?
n! multiplies every whole number from n down to 1. 5! = 120, and 0! is defined as 1.
Why is nCr smaller than nPr?
Each group of r items can be arranged in r! orders. Combinations count each group once.
What if r is bigger than n?
Without repetition there are no ways to choose, so the result is 0.