Permutations Calculator
Calculate the number of permutations P(n, r) — arranging r items from n where order matters. See also Combinations Calculator and Factorial Calculator.
How to Calculate Permutations
A permutation is an arrangement of items where order matters. To calculate P(n, r), divide n! by (n−r)!. For example, arranging 3 people in a line from a group of 10 — the order they stand in matters. The first position has 10 choices, the second has 9, and the third has 8, giving 10 × 9 × 8 = 720 permutations.
Permutations Formula
P(n, r) = n! / (n − r)!
Alternatively:
P(n, r) = n × (n-1) × (n-2) × ... × (n-r+1)
Relationship to combinations:
P(n, r) = C(n, r) × r!
Example
P(10, 3) = 10! / 7!
= 10 × 9 × 8
= 720
Compare with C(10, 3) = 120
P(10, 3) = C(10, 3) × 3! = 120 × 6 = 720 ✓
Frequently Asked Questions
When should I use permutations vs combinations?
Use permutations when the order of selection matters (rankings, passwords, seating arrangements). Use combinations when order does not matter (committees, lottery numbers, card hands).
What is P(n, n)?
P(n, n) = n!, which is the number of ways to arrange all n items. For example, P(4, 4) = 4! = 24 ways to arrange 4 items in a line.
What is P(n, 1)?
P(n, 1) = n. There are n ways to choose 1 item from n items when order matters (which is the same as combinations in this case).
How are permutations used in real life?
Permutations are used in password security (counting possible passwords), sports (race finishing orders), scheduling (task ordering), and cryptography.