Permutation and Combination Calculator

Count ordered arrangements or unordered selections, and compare what changes when repeated choices are allowed.

Printed from Calxy · https://www.calxy.net/math/permutation-combination-calculator

Total items (n)

Items chosen (r)

Combinations C(10, 3)

120

Permutations P(10, 3) = 720

Permutations, no repetition — nPr = n! ÷ (n − r)!720
Combinations, no repetition — nCr = n! ÷ (r!(n − r)!)120
Permutations with repetition — n^r1,000
Combinations with repetition — C(n + r − 1, r)220
r! (arrangements of each chosen set)6

All r for n = 10
r	nPr	nCr
0	1	1
1	10	10
2	90	45
3	720	120
4	5,040	210
5	30,240	252
6	151,200	210
7	604,800	120
8	1,814,400	45
9	3,628,800	10
10	3,628,800	1

How this was calculated

P(10, 3) = 10! ÷ 7! = 10 × 9 × 8 = 720

C(10, 3) = P(10, 3) ÷ 3! = 120; each combination can be ordered 3! ways.

Does order matter?

Permutations count ordered arrangements: choosing A then B differs from B then A. Combinations count unordered selections, so those choices are the same. Without repetition, P(n, r) = n! ÷ (n − r)! and C(n, r) = n! ÷ [r!(n − r)!].

With repetition allowed, ordered choices give n^r. Unordered choices give C(n + r − 1, r) for n > 0. These formulas assume distinct item types and no restrictions beyond the chosen repetition rule.

Example: choose 3 from 10

Without repetition, there are 10 × 9 × 8 = 720 ordered arrangements. Each group of three has 3! = 6 orders, so there are 720 ÷ 6 = 120 combinations.

With repetition, ordered choices give 10³ = 1,000. Unordered choices give C(12, 3) = 220.

Empty selections and large counts

Choosing zero items has one outcome: the empty selection. Choosing more items than are available has zero outcomes without repetition. From zero item types, choosing a positive number of items is impossible even with repetition.

Enter whole numbers n and r between 0 and 1,000. Arithmetic uses big integers; results longer than 24 digits are displayed in abbreviated scientific notation. That display is approximate even though the underlying count is exact.

Frequently asked questions

When should I use permutations?

Use permutations when position or sequence changes the outcome, such as assigning first, second and third prizes. Use combinations for a committee where membership matters but order does not.

What does repetition mean?

The same item type may be chosen more than once. A three-digit code using ten possible digits has 10³ ordered outcomes when repeated digits are allowed.

Last updated September 10, 2026. Results are estimates for informational purposes only.