Permutation & Combination Calculator

Count ordered permutations and unordered combinations for n objects chosen r at a time.

Enter your values

Change any field. The result is recalculated on this device.

Formula, example and assumptions
Combinations (nCr)120

Your result

Local calculation
Combinations (nCr)120

Combinations count selections where order does not matter; permutations count arrangements where it does. Both choose r objects from n distinct objects without replacement.

Total objects (n)
10
Objects chosen (r)
3
See how this is calculated

Result breakdown

Permutations (nPr)
720
Check the inputs and method

These are your current inputs, not the worked example further down the page.

Total objects (n)
10
Objects chosen (r)
3

Method

nPr = n!/(n − r)!; nCr = n!/[r!(n − r)!]. n! multiplies the integers 1 through n, with 0! = 1.

Read the worked example and assumptions

How this calculation works

Count selections from distinct objects without replacement. A permutation counts different orders separately, while a combination treats the same chosen group as one selection. Use combinations for a committee and permutations for ordered positions such as first, second and third.

nPr = n!/(n − r)!; nCr = n!/[r!(n − r)!]. n! multiplies the integers 1 through n, with 0! = 1.

Worked example

This example uses illustrative inputs. Your current result is shown above.

Selecting 3 from 10 distinct objects gives 10P3 = 10 × 9 × 8 = 720 ordered selections. Divide by 3! = 6 to ignore order: 10C3 = 720/6 = 120 groups.

Assumptions and limits

  • n and r are whole numbers with 0 ≤ r ≤ n ≤ 5,000.
  • Objects are distinct and cannot be selected more than once.
  • These counts do not handle repeated identical objects or replacement; large exact counts are displayed as integer text.

Questions people ask

How do I decide whether order matters?

Ask whether swapping two selected items creates a different outcome. Swapping committee members does not; swapping first and second prize winners does.

Why is choosing zero objects counted as one?

There is exactly one empty selection. Both formulas therefore return one when r is zero, including when n is also zero.

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