Permutations Formula
Calculate the number of ordered arrangements with P(n,r) = n! / (n-r)!.
Learn when order matters in counting problems.
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