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Choose k from n: Binomial Count

Count unordered selections of k distinct objects from n, without replacement.

  • Formula & worked example
  • Private in your browser
  • No signup
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Choosing two objects from five gives 10 unordered selections.

C(n,k) = n! / (k!(n − k)!)

Worked example

Enter these known values and leave the other values blank.

Available distinct objects n
5
Selected objects k
2
Result
10

Assumptions and limitations

  • Inputs are integers with 0 ≤ k ≤ n ≤ 20,000. The bound keeps evaluation responsive.
  • Distinct types are labeled A, B, C and so on in examples. Replacement permits repeated use of a type; ordering treats different position sequences as different outcomes.
  • Arithmetic uses 64 significant decimal digits. Large counts can be rounded, and scientific notation is approximate; this tool does not certify every digit of an arbitrarily large integer.

Common questions

Do order and replacement change the answer?

Yes. Order makes AB and BA different; replacement allows outcomes such as AA. Choose the model that matches both rules.

References

Bookify includes the valid empty-selection boundary while preserving the captured counting equations and enumeration ranges. The object count is bounded at 20,000.

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