How to use the Permutation Calculator
Enter n, the total number of items, and r, how many positions you are filling. The result counts every ordered arrangement.
Worked examples
Gold, silver and bronze among 10 runners can be awarded in 10 × 9 × 8 = 720 ways. A 4 digit PIN with no repeated digits has 10P4 = 5,040 options. If digits may repeat, the count is simply 10 × 10 × 10 × 10 = 10,000, which this formula does not cover.
Permutations and combinations
A permutation counts arrangements where order matters, while a combination counts groups where it does not. Every group of r items can be ordered in r! ways, so permutations equal combinations times r!. For 10 runners and 3 medals, there are 120 groups of three, each ordered 6 ways, giving 720.
The formula
nPr = n! ÷ (n − r)!
Examples
- P(10, 3)
P(10, 3) = 720
- P(8, 8), all arrangements
P(8, 8) = 40,320
- P(26, 4) four letter codes
P(26, 4) = 358,800
Limitations and assumptions
Exact integer arithmetic.
Frequently asked questions
What is the permutation formula?
nPr = n! ÷ (n − r)!.
Permutation or combination?
Use a permutation when order matters, such as podium places or PIN codes.
What if items can repeat?
Then the count is n^r instead.
How do they relate?
nPr = nCr × r!.