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Math

Permutations Calculator

Count ordered arrangements of k distinct items chosen from n 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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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, positions 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
Positions kNumeric value. Zero or positive. Maximum: 500.3

Result for these sample inputs

720
Order matters
Yes
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

P(n,k) = n! ÷ (n − k)!. Each change in order counts as a different arrangement.

A WORKED EXAMPLE

Filling 3 ordered positions from 10 distinct items gives 720 permutations.

What to keep in mind

Inputs must satisfy 0 ≤ k ≤ n ≤ 500. This excludes replacement and does not adjust for repeated identical objects.

Common questions

How is this different from combinations?

Permutations count order. Combinations count only which items are selected.

What if every item is arranged?

Setting k = n gives n!, the factorial of the number of distinct items.

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