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 − r)\!
Combinations (order doesn't matter):
nCr = n\! / [r\! × (n − 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 − 3)! = 10! / 7! = 10 × 9 × 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! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 ways
The committee has 6× fewer possibilities than the seating arrangement—because with a committee, {Alice, Bob, Carol} is the same as {Carol, Bob, Alice}.
When to Use Each
| Scenario | Method |
|---|---|
| Top-3 finishers in a race | Permutation |
| Choosing a 4-person team | Combination |
| PIN codes | Permutation |
| Lottery numbers | Combination |
| Password (alphabetical) | Permutation |
Factorial Shortcut
n! = n × (n−1) × (n−2) × ... × 1 0! = 1 (by definition) 5! = 5 × 4 × 3 × 2 × 1 = 120
Use our permutations and combinations calculator for any n and r.