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Unordered Selections with Replacement: Count & List

Count multisets of r choices from n distinct types and display the full list for small examples.

  • Formula & worked example
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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

Selecting two objects from three types with replacement gives six multisets.

C(n + r − 1,r) = (n+r−1)! / (r!(n−1)!)

Worked example

Enter these known values and leave the other values blank.

Available distinct types n
3
Selection size r
2
Combinations with repetitions
6

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. The captured n=4, r=4 outcome list omitted 15 of 35 multisets; Bookify supplies the complete independently enumerated list.

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