Combination Calculator
Combinations and Permutations Calculator works out Number of arrangements and Exact count from What to count, Total number of items (n) and Number chosen (k or r). The default figures on this page give number of arrangements of 2,598,960; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
2 steps
- Order does not matter and items are not reused, so this is the binomial coefficient.
- C(52,5) = 52! / (5! x 47!) = 2598960
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
2 sources
What the Combinations and Permutations Calculator calculates
Counts combinations and permutations of n items taken k at a time, with or without repetition, plus circular permutations and permutations of a multiset such as a word with repeated letters.
How to use this Combinations and Permutations Calculator
Enter What to count, Total number of items (n), Number chosen (k or r) and Repeat counts of each distinct item. Number of arrangements updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Number of arrangements, Exact count and Formula.
The method follows NIST Digital Library of Mathematical Functions, 26.3 Lattice Paths: Binomial Coefficients and 26.4 Multinomial Coefficients and Rosen, Discrete Mathematics and Its Applications, Chapter 6 (Counting): Theorems on permutations, combinations, and combinations with repetition.
Five-card poker hands from a 52-card deck example
With What to count: combination, Total number of items (n): 52, Number chosen (k or r): 5, the calculator returns Number of arrangements: 2598960, Exact count: 2598960. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Number of arrangements, Exact count and Formula. The inputs you can vary are What to count, Total number of items (n), Number chosen (k or r) and Repeat counts of each distinct item.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
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Frequently asked questions
What are the possible combinations of 1, 2, 3, 4, 5?
On the figures this page opens with, the number of arrangements is 2,598,960. No single number fits everyone: it follows What to count, Total number of items (n) and Number chosen (k or r), so enter your own above and the answer updates as you type.
What is 4 combination 2?
Counts combinations and permutations of n items taken k at a time, with or without repetition, plus circular permutations and permutations of a multiset such as a word with repeated letters. On this page it is computed from What to count, Total number of items (n) and Number chosen (k or r) and shown as Number of arrangements, Exact count and Formula.
What information do I need for the Combinations and Permutations Calculator?
You need What to count, Total number of items (n), Number chosen (k or r) and Repeat counts of each distinct item. The page opens with default figures that give number of arrangements of 2,598,960, so you can see the scale before entering your own.
What does this Combinations and Permutations Calculator show?
It shows Number of arrangements, Exact count, Formula, Combinations nCr, Permutations nPr, Combinations with repetition, Permutations with repetition (n^k) and Probability of one specific arrangement, with the working and any warnings for your entries.
Can I check how the Combinations and Permutations Calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.