Permutation Calculator
Permutation Calculator P(n, r)
About Permutation Calculator
What Is a Permutation?
A permutation is an ordered arrangement of objects. The key word is ordered: the arrangement ABC is different from BCA even though they use the same letters. Permutations answer questions like: how many different 3-letter codes can be made from 5 letters (without repetition)? The answer is P(5,3) = 5!/(5-3)! = 60. Permutations are used in password security (estimating possible passwords), tournament scheduling, DNA sequencing, and any situation where the order of selection matters.
The Permutation Formula
The formula for permutations of r items chosen from n items is P(n,r) = n! / (n-r)!. This equals n x (n-1) x (n-2) x ... x (n-r+1), which is the product of r consecutive integers starting from n and counting down. For permutations with repetition allowed, the formula is simply n^r (n raised to the power r).
How to Use This Calculator
Enter n (total objects) and r (objects to arrange). Select whether repetition is allowed. Click Calculate to see the number of permutations. The calculator also shows the step-by-step expansion so you can verify the math manually.
Frequently Asked Questions
What is the difference between permutation and combination?
Permutations care about order (ABC differs from BCA). Combinations do not (ABC and BCA are the same group). Permutations always give a larger or equal number than combinations for the same n and r.
When does order matter?
Order matters when positions are distinct: passwords, race finishing positions, phone numbers, license plates, rankings, and seating arrangements. Order does not matter for teams, committees, or groups.
What if items can repeat?
With repetition, the formula becomes n^r. A 4-digit PIN using digits 0-9 has 10^4 = 10,000 possibilities because each position can reuse any digit.
Related Tools
