Permutations Calculator
Count how many ordered arrangements of k items you can make from n items, written nPr. Order matters here, so it fits race finishes, codes with no repeated digits and seating plans.
- Runs in your browser
- Free, no account needed
- Copy or download the result
Result
Example
10
720
How to use the permutations calculator
- Type the total number of distinct items, n, in the box, from 0 through 10,000.
- Set Items to arrange, k (0 to 1,000). It must not be more than n.
- Press Run to see the number of permutations.
- Press Copy result or Download to save the full result.
How it works
Permutations = n! / (n - k)!, which is the same as multiplying n x (n - 1) x (n - 2) and so on for k terms.
Example: arranging 3 of 10 items gives 10 x 9 x 8 = 720.
Each item is used at most once, which is called without replacement. Codes where digits can repeat follow n^k instead, which this tool does not compute.
The answer is an exact whole number, computed with big integers rather than floating point.
n above 10,000, k above 1,000, negative numbers and k larger than n are rejected with an error.
Everything runs in your browser. Your input is not uploaded or stored.
Questions and answers
What is the formula for permutations?
P(n, k) = n! / (n - k)!. For 10 items taken 3 at a time, that is 10 x 9 x 8 = 720.
When do I use permutations instead of combinations?
Use permutations when order matters, such as first, second and third place. Use combinations when you only care which items are picked.
How many ways can 8 runners finish first, second and third?
P(8, 3) = 8 x 7 x 6 = 336.
What happens when k equals n?
You are arranging all n items, so the answer is n factorial. For n = 5 and k = 5, there are 120 orders.