How to Calculate Permutations and Combinations

Permutations and combinations are counting techniques that determine how many ways you can select or arrange items from a set. The key distinction: permutations care about order; combinations do not.

The Formulas

Permutations (order matters):

nPr = n\! / (n βˆ’ r)\!

Combinations (order doesn't matter):

nCr = n\! / [r\! Γ— (n βˆ’ r)\!]

Where n = total items, r = items chosen, ! = factorial.

Step-by-Step Examples

Permutation Example

How many ways can 3 students be arranged in 3 seats from a class of 10?

nPr = 10! / (10 βˆ’ 3)! = 10! / 7! = 10 Γ— 9 Γ— 8 = 720 ways

Combination Example

How many ways can 3 students be chosen for a committee from 10 (order doesn't matter)?

nCr = 10! / (3! Γ— 7!) = (10 Γ— 9 Γ— 8) / (3 Γ— 2 Γ— 1) = 720 / 6 = 120 ways

The committee has 6Γ— fewer possibilities than the seating arrangementβ€”because with a committee, {Alice, Bob, Carol} is the same as {Carol, Bob, Alice}.

When to Use Each

ScenarioMethod
Top-3 finishers in a racePermutation
Choosing a 4-person teamCombination
PIN codesPermutation
Lottery numbersCombination
Password (alphabetical)Permutation

Factorial Shortcut

n! = n Γ— (nβˆ’1) Γ— (nβˆ’2) Γ— ... Γ— 1 0! = 1 (by definition) 5! = 5 Γ— 4 Γ— 3 Γ— 2 Γ— 1 = 120

Use our permutations and combinations calculator for any n and r.