Permutation Calculator
Example Data Table
Use these sample cases to verify inputs and understand when each permutation model applies.
| Scenario | Mode | Inputs | Formula | Result |
|---|---|---|---|---|
| Award podium from 8 finalists | nPr | n = 8, r = 3 | 8! / 5! | 336 |
| Four-digit code using 10 digits | With repetition | n = 10, r = 4 | 10^4 | 10,000 |
| Round table seating for 7 guests | Circular | n = 7 | 6! | 720 |
| Arrange letters in BALLOON | Multiset | groups = 1,1,1,2,2 | 7! / (2! x 2!) | 1,260 |
Formula Used
Without repetition: P(n,r) = n! / (n-r)!
With repetition: P = n^r
Circular permutation: P = (n-1)!
Multiset permutation: P = n! / (a!b!c!...)
Permutation formulas count ordered outcomes. Order matters, so different positions create different arrangements. Factorials multiply descending integers, while duplicate groups divide repeated counts to avoid overcounting identical arrangements.
How to Use This Calculator
- Select the permutation type that matches your problem.
- Enter n for total items, then enter r if the mode needs positions.
- For repeated symbols, type duplicate group sizes separated by commas.
- Press Calculate Permutation to display the result above the form.
- Use Download CSV for result data or Download PDF to save a print-ready page.
Frequently Asked Questions
1. What is a permutation?
A permutation is an ordered arrangement of items. If the order changes, the arrangement changes too. That is why permutations differ from combinations.
2. When should I use nPr?
Use nPr when you choose r items from n distinct items and the order matters. Ranking winners, assigning seats, or making ordered selections are common examples.
3. What changes when repetition is allowed?
When repetition is allowed, each position can use any of the n items again. The count becomes n^r because every slot has the same number of choices.
4. Why does circular permutation use (n-1)!?
In a circle, rotating everyone together does not create a new arrangement. Fixing one item removes identical rotations, leaving (n-1)! distinct seatings.
5. What is a multiset permutation?
A multiset permutation handles repeated items, such as letters in a word. You divide by factorials of duplicate counts so identical swaps are not counted repeatedly.
6. Can this calculator handle large values?
Yes. It builds exact integers for values up to 500 using string arithmetic and factorial prime exponents, so results remain precise instead of rounding large counts.
7. Why are my inputs rejected?
The calculator checks valid ranges and relationships. For example, nPr requires r less than or equal to n, and duplicate groups must be comma-separated whole numbers.
8. Does the PDF button create a file directly?
The PDF button opens your browser print dialog. From there, choose your system's Save as PDF option to export the calculator page cleanly.