Probability Permutation Calculator

Calculator Form

Calculation mode

Total items or options (n)

Selected positions (r)

Repeated group counts

Favorable ordered outcomes

Use the repeated-item arrangement mode for words or objects with duplicates. Example: 3,2,2 means one group appears three times, and two groups appear twice.

Example Data Table

Scenario	Inputs	Formula	Outcome	Custom Probability
Without repetition	n = 8, r = 3, favorable = 4	8P3 = 8! / 5!	336	4 / 336 = 1.19047619%
With repetition	n = 5, r = 4, favorable = 10	5^4	625	10 / 625 = 1.6%
Circular permutation	n = 7, favorable = 3	(7 - 1)!	720	3 / 720 = 0.41666667%
Repeated-item arrangement	n = 7, groups = 3,2,2, favorable = 6	7! / (3! × 2! × 2!)	210	6 / 210 = 2.85714286%

Formula Used

1. Ordered selection without repetition: P(n, r) = n! / (n - r)!

2. Ordered selection with repetition: total outcomes = n^r

3. Circular permutation: total outcomes = (n - 1)!

4. Repeated-item arrangement: total outcomes = n! / (a! × b! × c! ...)

5. Probability of favorable ordered outcomes: probability = favorable outcomes / total outcomes

6. Exact one-order probability: probability = 1 / total outcomes

How to Use This Calculator

Select the permutation mode that matches your problem type.
Enter total items, positions, or repeated group counts as needed.
Optionally add favorable outcomes to calculate a custom probability.
Press the calculate button to show the result below the header.
Review the table, compare metrics in the graph, and export the result.

FAQs

1. What does this calculator measure?

It measures ordered arrangement counts and related probabilities. You can analyze standard permutations, repeated selections, circular arrangements, and repeated-item arrangements from one page.

2. When should I use the without repetition mode?

Use it when each item can appear only once in an ordered result. Typical examples include race finishing positions, lock codes without reuse, or ranked selections.

3. What is the difference between permutation and combination?

Permutation cares about order, while combination ignores order. This calculator shows combination count in the standard ordered-selection mode so you can compare both ideas directly.

4. Why does circular permutation use (n - 1)!?

A circular arrangement treats simple rotations as identical. Fixing one item removes duplicate rotations, leaving only the remaining items to be arranged.

5. How do repeated group counts work?

Enter counts like 3,2,2 when some symbols repeat. The calculator divides by each repeated factorial so identical swaps are not counted multiple times.

6. What does favorable outcomes mean?

Favorable outcomes represent the ordered results you want to count as success. The calculator divides that number by total outcomes to find your custom probability.

7. Why does the graph use log10 values?

Permutation counts can become huge while probabilities become tiny. A log10 graph keeps both visible, so comparisons remain readable on one chart.

8. Can I export the result for reports or study notes?

Yes. Use the CSV button for spreadsheet work and the PDF button for printable summaries. Both export the current result table shown above the form.