Permutation and combination calculator

How many ways to choose r items from n, when order does or doesn’t matter.

Formula Maths

Combinations nCr

Permutations nPr: 720
Combinations with repetition: 220
Permutations with repetition (nʳ): 1000

How it works

A permutation counts ordered selections: choosing a gold, silver and bronze winner. A combination counts unordered selections: choosing a team of three. Each combination corresponds to r! permutations.

With repetition, each pick can reuse an item: a 3-digit code from 10 digits has 10³ orderings. Unordered selections with repetition are counted with the stars-and-bars formula.

Formula

nPr = n! ÷ (n − r)!
nCr = n! ÷ (r! (n − r)!)
with repetition: nʳ and (n + r − 1)Cr

Worked example

Choosing 3 of 10: nPr = 10 × 9 × 8 = 720 ordered ways and nCr = 720 ÷ 3! = 120 groups. With repetition, there are 10³ = 1000 ordered and 12C3 = 220 unordered ways.

Questions people ask

What is the difference between a permutation and a combination?

Order matters in a permutation (ABC differs from CBA) but not in a combination (ABC and CBA are the same group).

How many lottery combinations are there for 6 from 49?

49C6 = 13,983,816, so one ticket has a 1 in 13,983,816 chance of matching all six.

What is n choose r?

Another name for nCr, also called the binomial coefficient and written (n r) in a tall bracket.