Count arrangements and selections: nPr and nCr.
nCr counts selections where order does not matter; nPr counts arrangements where it does. Choosing 3 from 10 gives 120 combinations or 720 permutations.
Permutations count the ways to arrange items when order matters; combinations count the ways to choose them when it does not. Enter n (total items) and r (items chosen) and the calculator returns nPr, nCr, the versions with repetition allowed, and n! and r!, using exact big-integer arithmetic so even C(100, 50) shows every digit.
Counting underlies probability — see the Probability Calculator — and appears in lotteries and passwords. Related: Big Number Calculator (factorials), Random Number Generator and Number Sequence Calculator.
The calculator applies the standard method step by step.
| Item | Value |
|---|---|
| Permutations nPr (order matters) | 720 |
| Combinations nCr (order does not matter) | 120 |
| Permutations with repetition (nʳ) | 1,000 |
| Combinations with repetition C(n + r − 1, r) | 220 |
| n! | 3,628,800 |
| r! | 6 |
| Formula check: nPr ÷ r! = nCr | confirmed |
| Case | nCr |
|---|---|
| n = 5, r = 2 | 10 |
| n = 10, r = 3 | 120 |
| n = 20, r = 5 | 15,504 |
| n = 52, r = 5 | 2,598,960 |
| n = 100, r = 50 | 100,891,344,545,564,193,334,812,497,256 |
| n | r = 0 | r = 1 | r = 2 | r = 3 | r = 4 | r = 5 |
|---|---|---|---|---|---|---|
| 0 | 1 | — | — | — | — | — |
| 1 | 1 | 1 | — | — | — | — |
| 2 | 1 | 2 | 1 | — | — | — |
| 3 | 1 | 3 | 3 | 1 | — | — |
| 4 | 1 | 4 | 6 | 4 | 1 | — |
| 5 | 1 | 5 | 10 | 10 | 5 | 1 |
| 6 | 1 | 6 | 15 | 20 | 15 | 6 |
| 7 | 1 | 7 | 21 | 35 | 35 | 21 |
| 8 | 1 | 8 | 28 | 56 | 70 | 56 |
| 9 | 1 | 9 | 36 | 84 | 126 | 126 |
| 10 | 1 | 10 | 45 | 120 | 210 | 252 |
In a permutation order matters (ABC ≠ CBA); in a combination it does not.
nCr = n! ÷ (r! × (n − r)!). C(5, 2) = 120 ÷ (2 × 6) = 10.
C(52, 5) = 2,598,960 five-card hands.
n! is the product of all positive integers up to n: 5! = 120. By definition 0! = 1.
Choosing r items from n types when the same type can be chosen more than once: C(n + r − 1, r).
The coefficients of (a + b)ⁿ are the values nC0, nC1 … nCn — the rows of Pascal's triangle.
It uses the standard method: nPr = n! ÷ (n − r)! nCr = n! ÷ (r! (n − r)!) The full working and a worked example are shown on this page.
Decimal results use standard double-precision arithmetic (about 15–16 significant digits). Tools that work with whole numbers — big numbers, factors, bases, combinations — use exact integer arithmetic, so no digits are lost.
Yes for checking your work and learning the method. The steps and formula are shown so you can understand the answer, but follow your teacher's rules about calculators in exams.
No. Everything is calculated in your browser and nothing you enter is uploaded or saved.
