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Math

Combinations Calculator

Count unordered selections of k distinct items from n items without replacement.

Exact method explained belowEditable inputs
Calculation notebook
Calculation notebook

The formula and worked example below explain the calculation.

See the method

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Your result

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Use the Calculate button to see your result.

Review the method below for assumptions and conventions.

How to use this tool

  1. Enter available items n, selected items k.
  2. Select Calculate to view the result.
  3. Check the method and assumptions below before using the result.

Input reference and sample values

Use the units and choices shown for each field. Sample values are provided for practice, not as recommendations.

InputFormat and unitsSample
Available items nNumeric value. Zero or positive. Maximum: 500.10
Selected items kNumeric value. Zero or positive. Maximum: 500.3

Result for these sample inputs

120
Order matters
No
Replacement
No

Select Try the example in the form to load these values and calculate them again. This replaces the current form inputs. Compare the result with the method and worked explanation below.

The method, explained

C(n,k) = n! ÷ [k!(n − k)!]. Equivalent selections in different orders count only once.

A WORKED EXAMPLE

Choosing 3 items from 10 gives 120 combinations.

What to keep in mind

Inputs are whole numbers with 0 ≤ k ≤ n ≤ 500. Items are distinct, and repetitions are not allowed.

Common questions

Does order matter?

No. Choosing A, B and C is the same selection as C, A and B.

What if I choose zero items?

There is one empty selection, so C(n,0) = 1.

Methodology maintained by ClarityKit. How these tools are built and checked.