Calculator Inputs
Example Data Table
| Case | Inputs | Formula | Result |
|---|---|---|---|
| Standard selection order | n = 10, r = 3 | 10P3 = 10! / 7! | 720 |
| Full arrangement | n = 6 | 6! | 720 |
| Repeated symbols allowed | n = 5, r = 4 | 5^4 | 625 |
| Circular ordering | n = 7 | (7 - 1)! | 720 |
| Multiset arrangement | counts = 2,2,1 | 5! / (2! × 2! × 1!) | 30 |
Formula Used
1. Standard permutation: nPr = n! / (n - r)! . This counts ordered selections of r items chosen from n distinct items.
2. Full permutation: n! . This counts every possible ordering of n distinct items.
3. Permutation with repetition: nr . Each position can reuse any available symbol.
4. Circular permutation: (n - 1)! . One item is fixed to remove rotational duplicates.
5. Circular permutation with mirror symmetry: (n - 1)! / 2 for n > 2. Clockwise and anticlockwise orders are treated as the same.
6. Multiset permutation: n! / (a! × b! × c! × …) . Repeated groups reduce the total number of unique arrangements.
How to Use This Calculator
- Select the permutation model that matches your counting problem.
- Enter n and r for standard, full, repeated, or circular cases.
- For multiset problems, enter repeated counts such as 2,2,1.
- Choose whether mirror-image circles should be considered identical.
- Enable sample arrangements if you want a small preview list.
- Press the calculate button to show the result above the form.
- Use the export buttons to download the report as CSV or PDF.
Frequently Asked Questions
1. What is the difference between combinations and permutations?
Permutations care about order, while combinations ignore order. For example, ABC and BAC are different permutations but the same combination.
2. When should I use standard nPr?
Use standard nPr when you select r items from n distinct items and the arrangement order matters, such as ranking winners or assigning ordered seats.
3. What does full permutation mean?
Full permutation means arranging every available distinct item. It is simply n!, where n is the number of unique items being ordered.
4. What is a permutation with repetition?
This mode counts ordered results when symbols can be reused. Password patterns, lock codes, and repeated digit strings commonly use n raised to r.
5. Why is circular permutation based on (n - 1)!?
Rotating a circular arrangement does not create a new unique order. Fixing one item removes these duplicates, leaving (n - 1)! distinct arrangements.
6. What is a multiset permutation?
A multiset permutation handles repeated items, such as letters in BALLOON. Repetition reduces the total because swapping identical items changes nothing.
7. Why does this calculator show very large exact integers?
The calculator uses digit-string arithmetic for exact counting. That avoids rounding errors and helps when factorial-based values exceed normal integer limits.
8. Why are some sample arrangements unavailable?
Preview generation is limited to smaller inputs so the page stays responsive. The exact count is still computed even when arrangement samples are hidden.