Calculator Inputs
Formula Used
If total distinct items are N, and grouped blocks have sizes g₁, g₂, ..., gₖ, then the grouped items stay together as blocks.
Let S = N - (g₁ + g₂ + ... + gₖ). These are the single items outside every group.
The number of units placed in line is B = S + k.
Block arrangements: B!
Duplicate-group adjustment: divide by m₁!m₂!... when same-sized groups are intentionally treated as identical.
Internal arrangement factor: g₁! × g₂! × ... × gₖ! when internal order is allowed.
Favorable grouped permutations:
F = (B! / duplicate divisor) × internal factor
Total unrestricted permutations: N!
Probability of that grouped pattern: P = F / N!
How to Use This Calculator
- Enter the total number of distinct items in the arrangement.
- Type group sizes as comma-separated integers, such as 3,2,2.
- Choose whether members inside each group may rearrange.
- Choose whether same-sized groups should be treated as identical.
- Click the solve button to view counts, probability, and the chart.
- Use the CSV or PDF buttons to export the computed results.
Example Data Table
| Total items | Group sizes | Internal order | Same-sized groups identical | Singles | Block units | Favorable permutations | Probability |
|---|---|---|---|---|---|---|---|
| 10 | 3, 2, 2 | Yes | Yes | 3 | 6 | 8,640 | 0.002380952381 |
| 8 | 2, 2 | No | No | 4 | 6 | 720 | 0.017857142857 |
| 9 | 4, 3 | Yes | No | 2 | 4 | 576 | 0.001587301587 |
Frequently Asked Questions
1. What does a grouped permutation mean?
It counts linear arrangements where chosen items must stay together as blocks. The calculator first arranges blocks and singles, then optionally multiplies internal group orders.
2. When should internal order be fixed?
Use fixed order when each group already has a locked sequence. That means the block moves as one unit without rearranging its members.
3. When should same-sized groups be identical?
Choose identical when blocks of equal size should not be distinguished from each other. The calculator then divides by repeated-group factorial terms.
4. Does this calculator handle circular permutations?
No. This page uses a linear arrangement model. For circular seating, the formula changes because one rotational degree is removed.
5. Are items assumed distinct?
Yes. The unrestricted comparison uses N! for distinct items. Identical-item multiset permutations require a different counting model.
6. Why does the probability become very small?
Grouped patterns are usually rare among all unrestricted arrangements. As total items increase, N! grows rapidly and makes the grouped event less likely.
7. Why are some values shown in scientific form?
Factorials grow extremely fast. Scientific notation keeps very large counts readable while preserving scale for interpretation and exporting.
8. What happens if group sizes exceed total items?
The calculator stops and shows a validation message. Grouped items cannot exceed the total number of available items.