Combination Calculator

Calculate combinations, nCr or “n choose r” — the number of ways to pick r items from n when order does not matter — exactly, with the formula and every step shown.

Combinations (nCr) —

How to Calculate Combinations

A combination counts selections — where the order of the chosen items does not matter.

1

Enter n and r

Type the total number of items n and how many you are choosing r. Both must be non-negative whole numbers, with r no larger than n.

2

The calculator applies the formula

It computes nCr = n! / (r! × (n − r)!). The result is exact, even for large numbers, using BigInt arithmetic and the symmetry nCr = nC(n−r) to keep it precise.

3

Read the result and steps

The number of selections appears first, with the working below. If the order of the chosen items does matter, use the Permutation Calculator instead.

Frequently Asked Questions

What is a combination?
A combination is a selection of items in which the order does not matter. nCr, read “n choose r”, counts how many different groups of r items you can pick from a set of n. For example, the number of two-person committees from a group of ten is a combination, because the committee is the same regardless of who was named first.
What is the difference between a combination and a permutation?
A combination counts selections where order does not matter; a permutation counts arrangements where it does. Picking a three-person team is a combination, but arranging three people in first, second and third place is a permutation. nCr is always less than or equal to nPr because each unordered combination corresponds to r! ordered permutations.
What is the combination formula?
The number of combinations of r items chosen from n is nCr = n! / (r! × (n − r)!). It divides the total ordered arrangements nPr by r! to remove the orderings that count as the same selection. When r is 0 or equal to n, nCr is 1.
Does this calculator handle large numbers exactly?
Yes. It uses BigInt arithmetic and chooses the smaller of r and n−r to keep the calculation short, so combination counts such as the 2,598,960 possible five-card poker hands are computed exactly with no rounding error.

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