Permutation Calculator
Calculate the number of ways to arrange r items from a set of n items where order matters. Uses the formula P(n,r) = n!/(n-r)!
This free online permutation calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Total number of items in the set.
Number of items to arrange.
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Permutation Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Permutation Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Permutation Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Permutation Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
The Permutation Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the number of ways to arrange r items from a set of n items where order matters. Uses the formula P(n,r) = n!/(n-r)! The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
About Permutation Calculator
The permutation calculator determines the number of ordered arrangements of r items selected from a total of n distinct items. Unlike combinations, permutations treat different orderings as distinct outcomes. This concept is central to problems involving rankings, passwords, license plates, seating arrangements, and race finishing orders. If you need to know how many ways 3 runners can finish first, second, and third out of 10 competitors, permutations give the answer. The formula is derived from the multiplication principle: the first position can be filled n ways, the second (n-1) ways, and so on down to (n-r+1) ways for the rth position.
The Math Behind It
Formula Reference
Permutation Formula
P(n, r) = n! / (n - r)!
Variables: n = total items; r = items arranged; ! = factorial
Worked Examples
Example 1: Race finishing order
In a race with 10 runners, how many possible ways can the gold, silver, and bronze medals be awarded?
There are 720 possible medal outcomes.
Example 2: PIN code possibilities
How many 4-digit PINs can be formed from digits 0-9 without repeating any digit?
There are 5,040 possible PINs without repeated digits.
Common Mistakes & Tips
- !Using permutations when order does not matter. If selecting a committee, use combinations instead.
- !Confusing permutations with repetition (n^r) with permutations without repetition (n!/(n-r)!).
- !Forgetting that r cannot exceed n in the standard permutation formula.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
What is the difference between permutation and combination?
A permutation counts ordered arrangements (ABC is different from CBA), while a combination counts unordered selections (ABC and CBA are the same group). P(n,r) = C(n,r) * r!.
What if items can be repeated?
If repetition is allowed, the number of arrangements is n^r. For example, a 4-digit PIN with repetition allowed has 10^4 = 10,000 possibilities.
How do I handle permutations of identical items?
Use the multinomial formula: n!/(n1!*n2!*...*nk!) where n1, n2, etc. are the counts of each identical group. For example, the word MISSISSIPPI has 11!/(1!*4!*4!*2!) = 34,650 distinct arrangements.