Free · Combinatorics calculators 🔀

Compare Combination Counts with and without Replacement

Compare unordered selection counts with and without reuse of each object type, with a selectable small-example list.

  • 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

From five types, two selections give 10 subsets without replacement or 15 multisets with replacement.

Without replacement: C(n,r); with replacement: C(n+r−1,r)

Worked example

Enter these known values and leave the other values blank.

Available distinct types n
5
Selection size r
2
Combinations
10
Combinations with repetitions
15

Assumptions and limitations

  • Inputs are integers. The retained upper bound is 10,000 for each input. A selection of zero objects has one outcome: the empty selection.
  • 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.
  • Counts are available beyond the small-example listing ranges. A list appears only where the complete enumeration fits the supported range.
  • At least one available type is required in replacement/combined modes. Without-replacement selections with r > n have zero outcomes in combined comparisons.

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. Each count now controls its own numeric-display threshold.

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