Stars and Bars Calculator -- Combinations with Repetition

Stars and bars is a combinatorial method for counting the ways to distribute $n$ identical objects into $k$ distinct bins. Objects are represented as stars and dividers between bins as bars. The formula is $C(n + k - 1, k - 1)$.

If every bin must receive at least one object, pre-place one object in each bin, then distribute the remaining $n - k$ objects freely. The formula becomes $C(n - 1, k - 1)$. For example, distributing 10 candies among 3 children (each gets at least one) gives $C(9, 2) = 36$ ways.

Stars and bars solves three equivalent problems: distributing identical objects into distinct bins, counting non-negative integer solutions to $x_1 + x_2 + \ldots + x_k = n$, and selecting $n$ items from $k$ types with repetition allowed (multiset selection).

Stars and bars is exactly the formula for combinations with repetition. Choosing $n$ items from $k$ types (repeats allowed) equals $C(n + k - 1, n)$. Both problems reduce to arranging $n$ stars and $k - 1$ bars in a row.

Not directly. For constraints like “no bin gets more than $m$ objects,” use inclusion-exclusion on top of the basic stars and bars formula. Subtract cases where one or more bins exceed the limit, add back over-counted cases, and so on.