Combinations with Repetition Calculator
Find how many ways you can choose r items from n types when repeats are allowed and order does not matter.
Ways to choose with repetition, order ignored
- 1
Stars + bars pool: n + r − 1
5 + 3 − 1 = 7 - 2
Multiset combinations C(pool, r)
C(7, 3) = 35The number of ways to pick r items from n types allowing repeats, order ignored.
How does this calculator work?
Combinations with repetition count how many ways you can pick r items from n types when repeats are allowed and order does not matter. The answer is C(n + r − 1, r) = (n + r − 1)! / (r! · (n − 1)!), computed here multiplicatively, with comparisons to ordinary combinations and ordered selections.
Formula
How this is calculated
A combination with repetition (a multiset of size r drawn from n types) counts selections where the same type may be picked more than once and the order of selection is irrelevant. Enter n, the number of distinct types, as a positive integer, and r, the number of items chosen, as a non-negative integer.
The count is given by the stars-and-bars formula C(n + r − 1, r) = (n + r − 1)! / (r! · (n − 1)!). The calculator evaluates the binomial coefficient multiplicatively — multiplying r ascending terms and dividing as it goes — which avoids computing large factorials directly and keeps intermediate values small. For comparison it also reports C(n, r), the count when repetition is not allowed, and nʳ, the count when order also matters (ordered selections with repetition).
Inputs are rounded to integers. n must be at least 1; r may be 0, in which case there is exactly one selection (the empty multiset). Because results grow combinatorially, very large n and r can exceed exact double-precision integer range, so treat extreme outputs as approximate.
Frequently asked questions
A normal combination C(n, r) chooses r distinct items with no repeats. With repetition, each type can be chosen multiple times, giving C(n + r − 1, r), which is always at least as large.
Imagine r identical stars placed into n type-bins separated by n − 1 bars. Each arrangement of stars and bars is one multiset, and counting those arrangements gives C(n + r − 1, r).
Choosing zero items yields exactly one outcome — the empty selection — so the result is 1 for any n.
Also known as
TG we-Calculate Editorial Team. (2026). Combinations with Repetition Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/combinations-with-repetition-calculator
TG we-Calculate Editorial Team. "Combinations with Repetition Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/combinations-with-repetition-calculator.
TG we-Calculate Editorial Team, "Combinations with Repetition Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/combinations-with-repetition-calculator
@misc{wecalculate_combinations_with_repetition_calculator, title = {Combinations with Repetition Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/combinations-with-repetition-calculator}}, year = {2026}, note = {TG we-Calculate} }
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