Combinations with Repetition Calculator
Calculate combinations with repetition (multiset coefficients) where repetition is allowed.
Calculate all variants of combinations and permutations
What Is the Combinations with Repetition Calculator?
The Combinations with Repetition Calculator computes the number of ways to choose r items from n distinct types when repetition is allowed and order does not matter. This is also called a multiset coefficient and is commonly used in combinatorics, probability theory, and counting problems.
Formula
How to Use
Enter the number of distinct item types (n) and how many items you want to choose (r). The calculator instantly displays the number of possible combinations with repetition, along with the intermediate factorial values.
Example Calculation
Choose 2 scoops of ice cream from 4 flavors (with repetition allowed): C(4+2−1, 2) = C(5, 2) = 5! / (2! × 3!) = 120 / (2 × 6) = 10 combinations. You could pick two of the same flavor or any two different flavors.
Understanding Combinations with Repetition
Combinations with repetition arise in many areas of mathematics and real life. Unlike standard combinations where each element can be chosen only once, multiset combinations allow any element to appear multiple times in a selection. The formula C(n+r−1, r) generalizes Pascal's triangle to handle these cases elegantly.
This type of counting problem frequently appears in probability distributions (e.g., occupancy problems), combinatorics coursework, and computer science algorithms. For example, counting the number of monomials of degree r in n variables, or distributing r identical objects into n distinct bins, both reduce to this formula.
The calculator handles the factorial arithmetic automatically, so you can solve problems involving large n and r without tedious manual computation. Results are exact integers, and the tool also shows intermediate values to help you understand each step of the calculation.
Frequently Asked Questions
What is the difference between combinations with and without repetition?
Without repetition (standard combinations), each item can be chosen at most once: C(n,r) = n!/(r!(n−r)!). With repetition, you can choose the same item multiple times: C(n+r−1, r).
Does order matter in combinations with repetition?
No. Combinations do not consider order. If you need ordered selections with repetition, that would be permutations with repetition: nʳ.
Can r be larger than n?
Yes. When repetition is allowed, r can exceed n because you can pick the same type more than once.
What are real-world examples of combinations with repetition?
Choosing pizza toppings (duplicates allowed), distributing identical candies into distinct bags, selecting repeated items from a menu, or counting polynomial terms.
Is this calculator free?
Yes, completely free with no account needed.