What is the difference between a combination and a permutation?

A combination ignores order — choosing A then B is the same as choosing B then A. A permutation treats order as significant, so AB and BA are counted separately. Use C(n,r) for combinations and P(n,r) = n!/(n−r)! for permutations.

What is C(n,0) or C(n,n)?

Both equal 1. There is exactly one way to choose zero items (choose nothing) and exactly one way to choose all n items (choose everything).

Why must r be less than or equal to n?

You cannot choose more items than exist in the set. If r > n there is no way to make the choice, so by convention C(n, r) = 0. The factorial formula does not apply because (n - r)! would be the factorial of a negative number, and this calculator shows an error instead.

How large can n and r be?

This calculator uses JavaScript BigInt, so every result is exact with no floating-point rounding. n can be at most 1000. The largest result, C(1000, 500), has 300 digits and is shown in full.

nCr is shorthand for "n choose r," the same as C(n,r) or the binomial coefficient. It appears on most scientific calculators as the Cn,r or nCr key.

How do I work out lottery odds with combinations?

Count the possible tickets with C(n, r). A 6-from-49 draw has C(49, 6) = 13,983,816 combinations, so one ticket has a 1 in 13,983,816 chance at the jackpot. Powerball draws 5 of 69 white balls and 1 of 26 red balls: C(69, 5) × 26 = 11,238,513 × 26 = 292,201,338.

What if an item can be chosen more than once?

This page assumes each item is picked at most once. When repeats are allowed, such as 3 scoops from 5 ice cream flavors, the count is C(n + r - 1, r) = C(7, 3) = 35. Use the combinations with repetition calculator for that case.

Combination Calculator

Enter n and r to instantly compute C(n,r) — the number of unordered combinations — with full factorial working shown.

C(n, r) = n! ÷ (r! × (n - r)!) counts the ways to choose r items from n when order does not matter.

Combination Calculator — C(n, r)

n — total number of items

r — number of items to choose

Quick Examples

Result

Enter n and r to see the result

Copied!

Quick answer

C(n, r) = n! ÷ (r! × (n - r)!) counts the ways to choose r items from n when order does not matter. Choosing 3 of 10 items gives 120, a 5-card hand from a 52-card deck gives 2,598,960, and 6 numbers from 49 give 13,983,816. Enter n and r to see the exact result with the factorial working.

Examples

n	r	C(n, r)
5	2	10
10	3	120
20	5	15,504
52	2	1,326
52	5	2,598,960
49	6	13,983,816

Enter n and r.

Summary

A combination C(n,r) counts the number of ways to select r items from a set of n distinct items when order does not matter. It is calculated as n! / (r! × (n−r)!). This calculator computes the result instantly and displays the full factorial breakdown so you can verify every step.

How it works

Enter n — the total number of items in the set.
Enter r — the number of items you want to choose.
The calculator applies the formula C(n,r) = n! / (r! × (n−r)!).
The result and a step-by-step factorial breakdown appear immediately.
Use the Copy button to copy the result to your clipboard.

Use cases

Calculate lottery odds (e.g. how many ways to pick 6 numbers from 49).
Count hand combinations in card games like poker.
Determine the number of possible teams from a group of players.
Solve combinatorics homework problems with shown working.
Compute binomial distribution coefficients for statistics.
Find the number of committee compositions from a candidate pool.
Evaluate sample sizes for experimental design.
Check answers for nCr calculations in probability courses.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu