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Combination Calculator

Calculate unordered combinations from n items selecting r.

Combination Calculator Inputs

Results computed instantly — your data never leaves your device.

Live Combination Calculator Results

Real-Time

Combinations C(n,r)

120

Order is not considered in the selection.

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How to Use the Combination Calculator

  1. 1

    Enter the total population n.

  2. 2

    Enter the selected count r.

  3. 3

    Review the unordered selection count.

  4. 4

    Confirm that r is between zero and n.

Formula & Mathematical Basis

C(n,r) = n! ÷ (r!(n−r)!).

Variable Key

n

Total items available

r

Items selected without regard to order

📝 The implementation uses a multiplicative form to reduce unnecessary intermediate factorial growth.

Step-by-Step Examples

1

Choose three from ten

Scenario: Use n = 10 and r = 3.

  1. 1.Identify the ten available items.
  2. 2.Select three positions without ordering them.
  3. 3.Apply C(10,3) = 10! ÷ (3!7!).
  4. 4.Obtain 120 combinations.
✅ There are 120 unordered selections.

Practical Use Cases

  • Count groups where order does not matter.
  • Support probability and sampling exercises.
  • Compare combinations with permutations.

Common Pitfalls

  • Using permutations when order is irrelevant.
  • Entering r greater than n.
  • Treating repeated items as distinct without checking the model.

Frequently Asked Questions

What is the difference from a permutation?

A combination ignores order, while a permutation counts different orders separately.

What happens when r is zero?

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

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