Permutations Calculator
A permutation is an arrangement of items where the order does matter. The permutations formula P(n,r) = n! / (n-r)! calculates the number of ways to arrange r items from n total items. For example, if you have 5 people and want to arrange 2 in a line to win 1st and 2nd place, the order matters: Alice 1st and Bob 2nd is different from Bob 1st and Alice 2nd. The permutations formula gives you 20 possible arrangements. When you arrange all items, P(n,n) = n! (n factorial). This calculator shows both the formula result and a step-by-step breakdown of the multiplication. A reference table of factorials is included for quick reference.
Formula
P(n,r) = n! / (n-r)!
P(n,r) = n * (n-1) * (n-2) * ... * (n-r+1)
Step-by-step calculation
Factorial reference table
| n | n! |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5,040 |
| 8 | 40,320 |
| 9 | 362,880 |
| 10 | 3,628,800 |
Permutations calculator: frequently asked questions
What is a permutation?
A permutation is an arrangement of items where the order does matter. For example, arranging 3 people in a line from a group of 5 is a permutation. The formula is P(n,r) = n! / (n-r)!, where n is the total number of items and r is the number arranged.
What is the difference between P(n,r) and P(n,n)?
P(n,r) counts arrangements of r items chosen from n total items. P(n,n) counts arrangements of all n items, which is simply n! (n factorial). For example, P(5,5) = 5! = 120.
Why is permutation different from combination?
In permutations, order matters: ABC is different from BAC. In combinations, order does not matter: ABC and BAC are the same selection. Permutations give larger numbers because they count more arrangements.
What is n!?
n! (n factorial) is the product of all positive integers from 1 to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. By definition, 0! = 1.
What does the step-by-step calculation show?
The step-by-step calculation breaks down P(n,r) = n * (n-1) * (n-2) * ... * (n-r+1). This shows the multiplication of r consecutive numbers starting from n and counting down.
Official sources
- Wolfram MathWorld: Permutation.
- Wikipedia: Permutation.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.