Combinations and Permutations Calculator

Math

Combinations & Permutations Calculator

Last updated: July 11, 2026

Calculate combinations C(n,r) and permutations P(n,r) with step-by-step formula explanations. Quickly find the number of ways to choose or arrange items from a set.

Total Items (n)

Items Chosen (r)

How to Use This Calculator

Select whether to calculate Combinations, Permutations, or Both.
Enter the total number of items (n) and the number to choose (r).
Click Calculate to see results with full formula breakdown and factorial expansion.

Formula

Combinations (order does not matter):

C(n, r) = n! / (r! × (n − r)!)

Permutations (order matters):

P(n, r) = n! / (n − r)!

Where n! = n × (n−1) × (n−2) × ... × 1 (factorial).

Examples

Lottery Odds: Choose 6 numbers from 49. C(49,6) = 13,983,816 possible combinations — that's your chance of winning the jackpot.

Committee Selection: Choose 3 people from 10 for a committee. C(10,3) = 120 ways. If each person has a distinct role (chair, secretary, treasurer), then P(10,3) = 720 ways.

Pizza Toppings: Choose 3 toppings from 8 available options. C(8,3) = 56 unique topping combinations.

Frequently Asked Questions

What is the difference between combinations and permutations?

Combinations count selections where order does not matter (e.g., choosing a team). Permutations count arrangements where order does matter (e.g., ranking winners).

Why does C(n,r) = C(n,n−r)?

Choosing r items from n is the same as choosing which n−r items to leave out. So C(10,3) = C(10,7) = 120.

What if n is very large?

For n > 1000, results may overflow. This calculator limits n and r to 1000 and uses scientific notation for large results.

Can r be zero?

Yes. C(n,0) = 1 — there is exactly one way to choose nothing. Similarly, P(n,0) = 1.