Combinations With Repetition Calculator

Calculate the number of ways to choose k items from n types when repetition is allowed and order does not matter.

Combinations With Repetition
Formula
Examples
FAQ

Combinations With Repetition

Combinations with repetition (also called multiset coefficients or "stars and bars") count the ways to choose k items from n types when each type can be selected multiple times and order does not matter. This is equivalent to distributing k identical objects into n distinct bins.

Examples include choosing scoops of ice cream (can repeat flavors), distributing identical candies among children, and selecting items from categories where duplicates are allowed.

Formula

C(n+k-1, k) = (n+k-1)! / (k!(n-1)!)

Also written as: ((n+k-1) choose k)

Examples

Scenario	n (types)	k (choose)	Result
3 scoops from 5 flavors	5	3	35
4 donuts from 6 types	6	4	126
2 dice values (unordered)	6	2	21

FAQ

Why is the formula C(n+k-1, k)?

This comes from the "stars and bars" technique. Imagine k stars (items) and n-1 bars (dividers between types). The total symbols are n+k-1, and you choose where to place k stars (or equivalently n-1 bars), giving C(n+k-1, k).

When do I use with vs without repetition?

Use WITH repetition when items can be reused (choosing multiple scoops of same flavor). Use WITHOUT when each item is unique and can only be selected once (drawing cards from a deck).