Combination with Repetition Calculator

Explore the possibilities of combinations with repetition. Enter the number of items and items to choose to calculate the combinations.

Result:

Number of combinations with repetition:

Understanding Combinations with Repetition

Combinations with repetition, also known as combinations with replacement, determine the number of ways to choose r items from a set of n distinct items, where repetition of items is allowed.

The formula for combinations with repetition is given by:

$$C(n+r-1, r) = \binom{n+r-1}{r} = \frac{(n+r-1)!}{r!(n-1)!}$$

Example:

Suppose you want to choose 2 scoops of ice cream (r=2) from 3 available flavors: vanilla, chocolate, and strawberry (n=3), and you can choose the same flavor twice. The possible combinations are:

  • Vanilla, Vanilla
  • Vanilla, Chocolate
  • Vanilla, Strawberry
  • Chocolate, Chocolate
  • Chocolate, Strawberry
  • Strawberry, Strawberry

Using the calculator with n=3 and r=2, you will find there are 6 combinations, which matches our example.

What are Combinations with Repetition?

Combinations with repetition are a way to count the number of selections of items from a set where you are allowed to choose the same item multiple times. Unlike regular combinations where each item can be chosen at most once, combinations with repetition allow for items to be repeated in a selection.

This concept is useful in various scenarios, such as when you are selecting items from a menu where you can order the same dish multiple times, or when distributing identical items into distinct containers.

Formula and Calculation

The formula to calculate combinations with repetition is given by C(n+r-1, r), where n is the number of types of items to choose from, and r is the number of items to choose. This formula can be expanded as:

$$C(n+r-1, r) = \frac{(n+r-1)!}{r!(n-1)!}$$

This calculator simplifies the process of computing this value by directly applying this formula. Simply input the total number of distinct items (n) and the number of items you wish to choose (r), and the calculator will provide the result.

Uses and Applications