Combination (nCr) Calculator

Compute the number of ways to choose r items from n without regard to order, with the related permutation and symmetry.

Input

Calculate nCr, the number of ways to choose r items from n distinct items without regard to order. The permutation nPr and the symmetry are shown together.

Total items n

Enter an integer of 0 or more

Items to choose r

Enter an integer up to n

Result

Combination nCr of choosing 3 from 10

120

Permutation nPr arranging 3 of 10

720

Symmetry nC(n minus r) choosing 7 of 10

120

Digits of nCr

By symmetry nCr equals nC(n minus r). Choosing many items gives the same result as choosing the items left behind.

Pascal triangle

Each value is the sum of the two values above it, and row n holds the binomial coefficients nCk.

n = 0	1
n = 1	1 1
n = 2	1 2 1
n = 3	1 3 3 1
n = 4	1 4 6 4 1
n = 5	1 5 10 10 5 1
n = 6	1 6 15 20 15 6 1

Combination nCr equals n! divided by ( r! times (n minus r)! ). Permutation nPr equals n! divided by (n minus r)!, and multiplying nCr by r! gives nPr.

How it works

A combination nCr is the number of ways to choose r items from n distinct items without regard to order, given by nCr = n! divided by ( r! times (n minus r)! ).
The value is built with an iterative calculation that alternates multiplication and division. Each step divides exactly, so BigInt yields an exact integer even for large results.
Because of the symmetry nCr equals nC(n minus r), choosing many items gives the same result as choosing the items left behind.
The permutation nPr counts ordered arrangements and equals n! divided by (n minus r)!. Multiplying a combination by the number of orderings r! gives the permutation.
Pascal triangle is a triangle where each value is the sum of the two values above it, and the kth entry in row n equals the binomial coefficient nCk.
When r is greater than n no selection is possible, so the result is undefined. A large upper bound on n is applied to keep the computation safe.

Reviews

Tell us what you think of this calculator.