Calculator
Choose a mode, enter your values, and calculate exact arrangements. Large results are handled exactly with whole-number output.
Example Data Table
These examples show common use cases for each permutation style.
| Mode | Example Input | Formula | Result | Typical Use |
|---|---|---|---|---|
| Linear permutation | n = 10, r = 4 | 10! / 6! | 5,040 | Ordered selections |
| Repeated-item group permutation | 3,2,2 | 7! / (3! × 2! × 2!) | 210 | Duplicate symbols |
| Circular permutation | n = 6 | (6-1)! | 120 | Round-table seating |
| Grouped block permutation | n = 8, blocks = 3,2 | 5! × 3! × 2! | 1,440 | Items staying together |
Formula Used
1. Linear permutation
P(n,r) = n! / (n-r)!
Use this when you choose r ordered positions from n distinct items. Order matters, and each chosen position is unique.
2. Repeated-item group permutation
Unique arrangements = n! / (a! × b! × c! ...)
Use this when some items repeat. The denominator removes overcounting caused by identical items inside each repeated group.
3. Circular permutation
Circular arrangements = (n-1)!
If mirror images count as the same, arrangements = (n-1)! / 2 for n > 2.
Use this when rotation does not create a new arrangement, such as people around a round table.
4. Grouped block permutation
External arrangements = (remaining singles + number of blocks)!
Total arrangements = external arrangements × internal block arrangements
Use this when selected items must stay together as blocks, and optionally count the order inside each block.
How to Use This Calculator
- Select the permutation mode that matches your problem.
- Enter your counts, positions, or block sizes.
- Click Calculate Now to show the result above the form.
- Review the exact count, scientific notation, digits, and log-based graph.
- Use the CSV or PDF buttons to save the current result.
Frequently Asked Questions
1. What is a group permutation?
A group permutation counts ordered arrangements when items may be selected, repeated, grouped, or placed in blocks. It helps compare different arrangement rules within one calculator.
2. When should I use linear permutation mode?
Use linear permutation mode when order matters and you place or choose distinct items in sequence, such as ranking winners or assigning ordered positions.
3. What does repeated-item mode solve?
It handles identical items. When some symbols or labels repeat, ordinary factorial counting overstates the total, so duplicate factorials must divide the result.
4. Why is circular permutation smaller than linear permutation?
Rotating a circular arrangement does not create a new outcome. Fixing one position removes repeated rotations, which is why the formula becomes (n-1)!.
5. What does the grouped block option mean?
It treats certain items as units that must stay together. You can also include internal order if each block can rearrange within itself.
6. Why does the chart use log10 values?
Permutation results grow very quickly. A log10 graph keeps huge counts readable and lets you compare numerator, divisor, and final outcome in one view.
7. Can I export the result for reports?
Yes. The calculator includes CSV export for tabular reuse and PDF export for quick sharing, documentation, or record keeping.
8. Is the output exact or rounded?
The main result is exact. Scientific notation, log values, and bit estimates are summary views added for readability and comparison.