Steps without the calculator
Represent the k selected units as stars and place n−1 separators to divide them among the labeled types. Count all positions for stars or separators.
Math calculator
Combinations with Repetition Calculator
Count unordered selections of k items from n types when repeats are allowed. The page traces how the inputs become repeated combinations before rounding.
Combinations with Repetition inputs
Enter the source values
A worked example
Selecting seven items from five types with reuse allowed gives C(11,7)=330 possible count profiles. This Combinations with Repetition example can be compared with stars-and-bars coefficient.
Situations that fit the model
It models scoops by flavor, nonnegative integer solutions, inventory bundles, and allocations of identical units among labeled groups.
Why this relationship works
Stars and bars converts a repeated unordered selection into k stars separated among n categories, giving C(n+k−1,k). Combinations with Repetition can also be compared with ordered reuse.
For this page, combinations with repetition is interpreted under the stated convention and input order.
Mistakes worth catching
Items of the same type are treated as indistinguishable and order is ignored. Upper limits on individual types invalidate the unrestricted stars-and-bars formula.
Changing only one field helps distinguish a data-entry problem from the intended behavior of combinations with repetition.
Meaning, scale, and reporting
Transfer the source entries one at a time under number of item types and selection size, keeping any units visible in your notes. This page can validate numerical ranges, but only the reader can confirm that each entry represents the intended quantity.
Increasing k expands the set of count profiles; adding a type creates another possible compartment, including an empty one. Trying a nearby input should follow this pattern; otherwise inspect the field assignment.
Attach units and the name repeated combinations whenever the answer leaves this page. A rounded value is suitable for presentation, while retained working digits are safer for a dependent calculation.
Permutations with repetition count ordered strings, while this page counts only the final quantity assigned to each type. That neighboring measure needs its own setup rather than a relabeled answer.
Copying only the decimal discards the setup. Pair the repeated combinations with number of item types and selection size and the unit convention so another reader can reconstruct its meaning.
An independent route for Combinations with Repetition
For a small Combinations with Repetition example using Number of item types, Selection size, solve once with the displayed formula and once through direct enumeration or a probability tree. Two methods that organize outcomes differently are less likely to share the same hidden assumption error.
For Combinations with Repetition, when the exact list becomes too large, retain a simpler identity as a check: complementary probabilities sum to one, a Pascal row is symmetric, and adjacent factorial results have a known ratio. State which identity was used beside the reported answer.
Questions about Combinations with Repetition
Can a type be selected zero times?
Yes.
Does selection order matter?
No.
What if each type has a maximum?
This unrestricted formula no longer suffices.
Why use n+k−1?
It counts k stars and n−1 separators.