How to Calculate Combinations nCr

Calculates the number of ways to choose r items from n, nCr = nPr ÷ r!, where the order does not matter.

Explanation

A combination counts the ways of choosing $r$ items from $n$ when the order does not matter.

_nC_r = \dfrac{_nP_r}{r!} = \dfrac{n!}{r!\,(n-r)!}

It is the permutation count divided by $r!$ , the number of ways to shuffle any chosen set.

Example

Choose a committee of 3 from 5 people. Nobody is chair, so the order is irrelevant.

_5C_3 = \dfrac{5 \times 4 \times 3}{3 \times 2 \times 1} = \dfrac{60}{6} = 10

There are 10 committees: the 60 ordered arrangements divided by the $3! = 6$ orderings of each trio.

Useful identities

$_nC_r = {}_nC_{n-r}$ — choosing the 3 who join is the same as choosing the 2 who do not
$_nC_0 = {}_nC_n = 1$ — there is one way to choose nothing, and one way to choose everything
$_nC_r = {}_{n-1}C_{r-1} + {}_{n-1}C_r$ — exactly the rule that builds Pascal's triangle

Where it is used

Lotteries, card hands, committees and any other "just pick some" counting problem. The same numbers are the coefficients in the binomial expansion of $(a+b)^n$ , which is why they appear inside the binomial distribution.