Permutation & Combination Calculator (nPr, nCr)
Calculate permutations (nPr) and combinations (nCr) for any n and r. Shows step-by-step formula and factorial working.
About this calculator
Permutations count arrangements where order matters, and combinations count selections where it does not. Enter n, the total number of items, and r, the number you choose, and you get nPr and nCr with the factorial working.
Use it for passwords and PINs, seating, team selection, lottery odds and probability questions.
How to use it
- Enter n, the total number of items.
- Enter r, the number chosen.
- Read nPr, nCr and the step-by-step working.
Formulas
nPr = n! ÷ (n − r)! and nCr = n! ÷ (r! × (n − r)!). Permutations count each ordering as different, so nPr = nCr × r!.
Worked example
n = 10, r = 3
| 10P3 | 720 |
| 10C3 | 120 |
| 10! | 3,628,800 |
| Lottery 6 from 49 (49C6) | 13,983,816 |
Tips and common mistakes
- Ask: does order matter? If yes use permutations; if not use combinations.
- nC0 = nCn = 1.
- nCr equals nC(n − r), so choose the smaller r to compute.
- Factorials grow very quickly: 20! is about 2.4 × 10¹⁸.
- For repeated selection with replacement the formulas change.
Frequently asked questions
What is the difference between permutation and combination?
Permutations count ordered arrangements; combinations count unordered groups.
What is nPr?
The number of ways to arrange r items out of n.
What is nCr?
The number of ways to choose r items from n without regard to order.
What is 5!?
120.
What is 0!?
1 by definition.
How many ways can 3 people be seated from 10?
720.