Permutation and Combination Calculator

How This Calculator Works

A permutation (nPr) counts how many ways to arrange r items out of n where order matters, while a combination (nCr) counts how many ways to choose r items out of n where order doesn’t matter. Enter a total number of items (n) and how many you’re choosing (r), and this calculator finds both related answers at once.

The Formulas

• Permutations: $P(\vA{n}, \vB{r}) = \frac{\vA{n}!}{(\vA{n} - \vB{r})!}$ — equivalently, the product of $\vB{r}$ descending numbers starting at $\vA{n}$: $\vA{n} \times (\vA{n}-1) \times \cdots \times (\vA{n}-\vB{r}+1)$.
• Combinations: $C(\vA{n}, \vB{r}) = \frac{P(\vA{n}, \vB{r})}{\vB{r}!}$ — the same permutations count, divided by the $\vB{r}!$ ways those same $\vB{r}$ items could themselves be reordered (since order doesn’t matter for a combination).

Worked Example

Choosing 3 items from a group of 10, where order matters (permutations):

$$P(10,3) = 10 \times 9 \times 8 = 720$$

The same choice where order does NOT matter (combinations):

$$C(10,3) = 720 \div 3! = 720 \div 6 = 120$$

A Real-World Check

The odds of matching all 6 numbers in a 6-from-49 lottery draw are 1 in C(49,6) — exactly 13,983,816 — since the order the numbers are drawn in doesn’t change whether you’ve won.