Calculator Input
Use a word, a token list, or direct repeated counts.
Example Data Table
| Example | Repeated counts | Formula | Distinct permutations |
|---|---|---|---|
| BALLOON | 2, 2, 1, 1, 1 | 7! / (2! × 2!) | 1260 |
| BANANA | 3, 2, 1 | 6! / (3! × 2!) | 60 |
| MISSISSIPPI | 4, 4, 2, 1 | 11! / (4! × 4! × 2!) | 34650 |
| ASSESSMENT | 4, 2, 1, 1, 1, 1 | 10! / (4! × 2!) | 75600 |
| BOOKKEEPER | 3, 2, 2, 1, 1, 1 | 10! / (3! × 2! × 2!) | 151200 |
Formula Used
For repeated elements, the distinct permutation count is:
P = n! / (n1! × n2! × ... × nk!)
n is the total number of items.
n1, n2, ..., nk are repeated group counts.
The numerator counts every ordering.
The denominator removes duplicates caused by identical items.
If all counts are 1, the result becomes n!.
How to Use This Calculator
- Enter a word or sequence to count repeated symbols.
- Or enter repeated counts directly as numbers or labels.
- Choose character mode or token mode for sequence input.
- Add an optional expected total for a quick check.
- Select scientific notation precision and the chart view.
- Press calculate to see the result above the form.
- Review the exact count, formula, probability, and chart.
- Use the export buttons to save CSV or PDF output.
Frequently Asked Questions
1. What does this calculator find?
It finds the number of distinct arrangements when some items repeat. This avoids overcounting identical arrangements created by repeated letters, symbols, or tokens.
2. Can I enter a word instead of counts?
Yes. Enter a word like BALLOON in character mode. The calculator counts each symbol automatically, then applies the repeated permutation formula.
3. What is token mode for?
Token mode is useful when each item has more than one character. For example, red, red, blue, blue, green counts repeated words instead of repeated letters.
4. Why do repeated counts reduce the answer?
Repeated items create duplicate orderings that look different during counting but are actually identical. Dividing by each repeated factorial removes that duplication.
5. What does the probability result mean?
It shows the chance that one specific arrangement is selected from all distinct arrangements. That value is always 1 divided by the distinct permutation count.
6. What does the Plotly graph compare?
The graph compares the numerator, denominator, and final result. You can view either logarithmic magnitude or digit counts for a cleaner large-number comparison.
7. Can I use labeled counts?
Yes. Inputs like A:3, B:2, C:2, D:1 work well. Labels help you track which repeated groups belong to each count.
8. Why is there a size limit on total items?
Exact factorial values grow very quickly. A practical limit keeps the page responsive while still returning exact integer results for advanced teaching and reporting work.