What is the difference between combinations and combinations with repetition?

Standard combinations C(n,r) count selections where each item can be chosen at most once. Combinations with repetition allow the same type to appear multiple times, giving C(n+r-1, r) possibilities, which is always ≥ C(n,r).

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

By the "stars and bars" argument, choosing r items from n types with repetition is equivalent to placing r stars among n-1 dividers, giving C(n+r-1, r) arrangements.

What happens when r = 0?

There is exactly 1 way to choose zero items (the empty selection), so the result is always 1 when r = 0, regardless of n.

Yes. With only one type, there is exactly 1 combination no matter how many items you choose, because all r items must be that single type.

How large can the inputs be?

This tool accepts n and r up to 300. Results beyond about 15 digits are shown in full using big-integer arithmetic, so no precision is lost.

Combinations With Repetition Calculator

Enter the number of types (n) and items to choose (r) to get the multiset coefficient C(n+r-1, r).

Inputs

n — number of distinct types

r — items to choose (repetition allowed)

Formula

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

Result

Enter n and r above, then click Calculate.

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Summary

Combinations with repetition count the number of ways to choose r items from n distinct types when order does not matter and the same type can be chosen multiple times. The formula is the multiset coefficient C(n+r-1, r) = (n+r-1)! / (r! × (n-1)!). This is also called "stars and bars" or multichoose. For example, choosing 3 scoops from 4 ice-cream flavors gives C(6,3) = 20 possible combinations.

How it works

Enter n — the number of distinct types to choose from.
Enter r — the number of items to select (repetition allowed).
The tool computes C(n+r-1, r) using exact integer arithmetic to avoid overflow.
The result is displayed with the full expanded formula and step-by-step values.
Copy the result to your clipboard with one click.

Use cases

Count ice-cream combinations when a flavor can be picked more than once.
Calculate the number of ways to distribute identical items among categories.
Solve "stars and bars" problems in combinatorics homework.
Determine possible outcomes when rolling a die r times and ignoring order.
Enumerate multisets of size r drawn from an alphabet of n symbols.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu