Permutation & Combination Calculator
Calculate nPr, nCr, factorials, circular arrangements, and repetition-based counting formulas with exact counts and compact large-number output.
Last Updated: May 2026
Number of distinct available items or categories.
Number of items chosen or ordered.
Combinations
2,598,960
Approximation
2.598960e+6
Digits
7
Formula
nCr = n! / (r!(n-r)!)
| Step | Value |
|---|---|
| Formula | nCr = n! / (r!(n-r)!) |
| Substitution | 52C5 = 52! / (5!(52-5)!) |
| Order matters? | No |
| Repetition allowed? | No |
| Interpretation | Choose 5 items from 52 when order does not matter. |
Exact result
2,598,960
Use permutations when ABC and BAC are different outcomes, such as rankings or codes.
Use combinations when ABC and BAC are the same group, such as teams or card hands.
Select a repetition mode when an item can be used more than once, such as PIN digits.
Decide order and repetition before calculating. Most mistakes in counting problems come from choosing nPr when nCr is needed, or treating repeated choices as if they were not allowed.
Counting Method Notice
This calculator is for educational and planning use. Counting results depend on the correct assumptions about distinct items, order, repetition, circular equivalence, and replacement rules.
Checked by Jitendra Kumar
Permutation & Combination Calculator is checked for formula labels, source links, and result limits.
Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.
How to Use the Permutation & Combination Calculator
Choose the counting method first. Use combinations when order does not matter, permutations when order matters, and repetition modes when an item can be reused.
Enter total items as n and selected items as r. Factorial and circular permutation modes use only n.
Step 1: Choose the counting method
Select nCr, nPr, repetition, factorial, or circular permutation mode.
Step 2: Enter n and r
Use n for total available items and r for how many are selected or arranged.
Step 3: Review the count
Check the exact result, approximation, digit count, and formula substitution.
Step 4: Confirm assumptions
Use the order/repetition notes to verify the selected counting model.
How This Permutation & Combination Calculator Works
The calculator applies the counting formula that matches the selected method. For standard combinations and permutations, it assumes distinct items and no repetition.
Repetition modes change the formula because selected items can be reused. Circular permutation mode treats rotations around a circle as the same arrangement, so it uses one fewer factorial.
Results are calculated as exact integers when practical. Very large outputs are shown with digit counts and scientific notation so the result stays readable.
Permutation and Combination Guide
Core Counting Formulas
| Method | Formula | Use |
|---|---|---|
| Combination | nCr = n! / (r!(n-r)!) | Order does not matter, no repetition. |
| Permutation | nPr = n! / (n-r)! | Order matters, no repetition. |
| Combination with repetition | C(n+r-1, r) | Order does not matter, repetition allowed. |
| Permutation with repetition | n^r | Order matters, repetition allowed. |
| Factorial | n! | Arrange all n distinct items in a line. |
| Circular permutation | (n - 1)! | Arrange n distinct items around a circle when rotations match. |
Choosing the Right Method
| Scenario | Order matters? | Repetition? | Use |
|---|---|---|---|
| Choosing a committee | No | No | Combination |
| Ranking top finishers | Yes | No | Permutation |
| Creating a PIN code | Yes | Yes | Permutation with repetition |
| Choosing scoops from flavors | No | Yes | Combination with repetition |
| Arranging people around a table | Yes, circular | No | Circular permutation |
The key question is whether a different order creates a different outcome. If a team of Alex, Blair, and Casey is the same team in any order, use a combination. If first, second, and third place are different outcomes, use a permutation.
Once you have a count, it can become the denominator of a probability problem. Use the probability calculator to turn favorable counts into event probabilities, complements, or binomial results.
Keep the research moving with Probability Calculator, Statistics Calculator, Fraction Calculator, and Scientific Calculator.
Frequently Asked Questions
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- 1.OpenStax Contemporary Mathematics - Permutations(Accessed May 2026)
- 2.OpenStax Contemporary Mathematics - Combinations(Accessed May 2026)
- 3.OpenStax Contemporary Mathematics - Formula Review(Accessed May 2026)