Permutations Formula
Permutations formula P(n,r) = n! / (n−r)! counts ordered arrangements of r items from n.
Learn when order matters versus combinations with worked examples.
The Formula
P(n, r) = n! / (n - r)!
The permutations formula counts the number of ways to arrange r items out of n total items.
Order matters in permutations — ABC is different from CBA.
Variables
| Symbol | Meaning |
|---|---|
| P(n, r) | Number of permutations of r items from n items |
| n | Total number of items available |
| r | Number of items being chosen/arranged |
| n! | n factorial |
Example 1
How many 3-letter arrangements can be made from the letters A, B, C, D, E?
n = 5 (five letters), r = 3 (choosing 3)
P(5, 3) = 5! / (5 - 3)! = 5! / 2!
P(5, 3) = 120 / 2
P(5, 3) = 60 arrangements
Example 2
In a race with 10 runners, how many ways can gold, silver, and bronze be awarded?
n = 10 (runners), r = 3 (medal positions)
P(10, 3) = 10! / (10 - 3)! = 10! / 7!
P(10, 3) = 10 × 9 × 8 = 720
P(10, 3) = 720 ways
When to Use It
Use the permutations formula when:
- The order of selection matters (1st, 2nd, 3rd place finishes)
- Arranging people in seats or positions
- Creating codes, passwords, or PIN numbers (without repetition)
- Scheduling tasks in a specific order
Key Notes
- Formula: P(n, r) = n! / (n − r)!: The number of ways to arrange r items chosen from n distinct items, where the order of selection matters. P(5, 3) = 5!/2! = 60.
- When r = n (full permutation): P(n, n) = n!, since (n − n)! = 0! = 1. For example, the number of ways to arrange 4 books in a row is 4! = 24.
- Permutations vs combinations: Permutations count ordered arrangements; combinations count unordered selections. P(n, r) = r! × C(n, r) — permutations are always ≥ combinations for the same n and r.
- Permutations with repetition: If repetition is allowed, the count is nʳ. A 4-digit PIN from digits 0–9 with repetition has 10⁴ = 10,000 possible values.
- Circular permutations: When arranging n objects in a circle, the count is (n − 1)! because rotations of the same arrangement are considered identical. Seating 6 people around a table gives 5! = 120 arrangements.