What is the formula for combinations with repetition?

The formula is C(n+r-1, r) = (n+r-1)! / (r! × (n-1)!), also written as the multiset coefficient ⟨n r⟩. Here n is the number of distinct types and r is how many you are choosing. The derivation comes from the stars-and-bars argument: placing r identical balls into n labelled bins is equivalent to arranging r stars and (n-1) bars in a row of n+r-1 symbols.

How is combinations with repetition different from standard C(n, r)?

Standard combinations C(n, r) count ways to pick r distinct items from n when no item can be repeated and order does not matter. Combinations with repetition allow the same type to be chosen multiple times, so the effective pool grows to n+r-1 positions. The result C(n+r-1, r) is always greater than or equal to C(n, r) for the same n and r.

What does stars and bars mean?

Stars and bars is a visual proof technique. Imagine r identical stars (the items you pick) and n-1 vertical bars (the dividers between the n types). Any arrangement of these r+n-1 symbols uniquely defines a multiset selection — the count of stars between consecutive bars tells you how many of each type you chose. The number of arrangements is C(n+r-1, r).

Can r be larger than n in this formula?

Yes — that is precisely the point of allowing repetition. You can pick r = 10 items from only n = 3 types because each type can appear multiple times. In contrast, standard combinations require r ≤ n. With repetition, C(3+10-1, 10) = C(12, 10) = 66.

When is the answer equal to 1?

The answer is 1 in two cases: when r = 0 (the only selection is the empty selection, regardless of n) and when n = 1 (only one type exists, so every selection of any size must consist entirely of that type — exactly one way).

What are real-world uses of combinations with repetition?

Common applications include counting flavour combinations at an ice-cream counter, the number of ways to distribute identical objects among labelled bins, generating multisets in probability problems, counting monomials of a given degree in polynomial algebra, and enumerating solutions to non-negative integer equations x1 + x2 + ... + xn = r.

Combinations with Repetition Calculator

Combinations with repetition — also called multiset coefficients — answer the question: “In how many ways can I choose r items from n types when the same type may be picked more than once and the order of picking does not matter?” The answer comes from one of combinatorics’ most elegant results: the stars-and-bars theorem.

How it works

The formula is:

C(n+r−1, r) = (n+r−1)! ÷ (r! × (n−1)!)

where n is the number of distinct item types available and r is the size of the selection you want to make. The trick is to reduce the problem to a standard binomial coefficient by imagining an expanded pool of n+r−1 positions.

Stars-and-bars derivation

Picture r identical stars (one per item chosen) and n−1 vertical bars (the dividers between the n types). Every way to arrange these n+r−1 symbols in a row corresponds to exactly one multiset selection: the number of stars between bars 0 and 1 says how many of type 1 you picked, between bars 1 and 2 how many of type 2, and so on.

Counting the arrangements is simply choosing which r of the n+r−1 positions hold stars — that is, the standard combination C(n+r−1, r). No stars in a gap means zero of that type, which is perfectly legal, so repetition is fully captured.

Worked example

Scenario: An ice-cream shop offers 5 flavours. You order 3 scoops — any flavour can appear more than once (e.g. triple chocolate is allowed). How many distinct orders exist (ignoring the order in which scoops are placed on the cone)?

n = 5 (flavours), r = 3 (scoops)
m = n + r − 1 = 5 + 3 − 1 = 7
Answer = C(7, 3) = 7! ÷ (3! × 4!) = 5040 ÷ (6 × 24) = 35

n (types)	r (choices)	C(n+r−1, r)	Standard C(n, r)
5	3	35	10
4	2	10	6
3	10	66	n/a (r > n)
10	5	2002	252

Notice how allowing repetition always produces a larger (or equal) count, and that repetition makes it valid for r to exceed n — something impossible in the standard case.

Formula note

The formula C(n+r−1, r) is sometimes written with the multiset notation ⟨n r⟩ or ((n, r)) to distinguish it from the standard C(n, r). In generating-function terms, it is the coefficient of x^r in the expansion of 1/(1−x)^n. In number theory, it counts the number of non-negative integer solutions to x₁ + x₂ + … + xₙ = r. All three perspectives are equivalent.

This calculator uses exact BigInt arithmetic, so the result is always a precise integer — there is no floating-point rounding even for very large n and r. Results with dozens or hundreds of digits are displayed correctly.