Permutation Calculator
Calculate permutations P(n,r) = n!/(n−r)! for ordered arrangements.
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Quick Tip
Enter n, the total number of items available.
About This Tool
Why use Permutation Calculator?
A permutation counts arrangements where the order matters: P(n, r) = n! ÷ (n − r)!. Choosing 3 finishers from 8 runners gives 8! ÷ 5! = 336 possible first, second and third placings.
The tool shows the permutation count along with n! and (n − r)! so the working is visible. Factorials grow extremely quickly — 20! is already about 2.4 × 10¹⁸ — so very large values lose precision and are reported as estimates rather than exact counts.
Key Benefits
Working shown
n! and (n − r)! are displayed next to the answer, which is what most exam questions ask you to write out.
Order-sensitive counting
Use this when first, second and third — or a password's letter order — are different outcomes.
Guards against invalid input
r cannot exceed n, and the tool says so rather than returning a meaningless number.
How to use Permutation Calculator
- 1
Enter n, the total number of items available.
- 2
Enter r, how many of them are chosen and arranged.
- 3
Read P(n, r), the number of ordered arrangements.
- 4
Compare with the combination count if order does not actually matter in your problem.
Best use cases
Count the possible finishing orders in a race.
Work out how many ways a committee can be arranged into ranked roles.
Estimate how many ordered codes a given alphabet can produce without repeats.
Check permutation questions in probability homework.
Permutation Calculator FAQs
What is the difference between a permutation and a combination?
Permutations count arrangements where order matters; combinations count selections where it does not. P(5, 3) = 60 while C(5, 3) = 10, because each group of 3 can be ordered in 3! = 6 ways.
What is the permutation formula?
P(n, r) = n! ÷ (n − r)!. For P(8, 3): 8! ÷ 5! = 40,320 ÷ 120 = 336.
What is 0! and why is it 1?
Zero factorial is defined as 1. It keeps formulas consistent: P(n, n) = n! ÷ 0! must equal n!, which only works if 0! = 1.
Does this handle permutations with repetition?
No. This is the no-repeat case. If items can repeat, the count is simply nʳ — for a 4-digit PIN from 10 digits, 10⁴ = 10,000.
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