Statistics Tool

Combinations with Replacement

Calculate the number of ways to select items from a set where each item can be chosen more than once.

Σ The Formula

C(n+r-1, r) = (n+r-1)! / [r!(n-1)!]

Real World Examples

Ice Cream
Choosing 3 scoops from 5 flavors where duplicates are allowed → 35 ways
Stars and Bars
Allocating identical rewards to distinct people.

# About This Calculator

Combinations with Replacement (also known as Multiset Coefficients or "Stars and Bars") count the ways to pick $r$ elements from $n$ categories where you can pick the same category multiple times.

Unlike standard combinations, the pool of items doesn't shrink after a selection. This is common in probability problems involving rolling multiple dice or selecting items from a stock that is replenished.

How To Use

  1. Enter the **Total Categories (n)** available to pick from.
  2. Enter the **Number of Selections (r)** you want to make.
  3. The tool calculates the total unique sets possible, ignoring the order of selection.

Frequently Asked Questions

What is 'Replacement'?+

It means that after you pick an item, you 'put it back' or consider it available again for the next pick. You can have multiples of the same item in your selection.

Why is it called Stars and Bars?+

In combinatorics, this problem is often visualized as placing $r$ identical 'stars' into $n$ 'bins' separated by $n-1$ 'bars'.

Is Combinations with Replacement free to use?+

Yes, Combinations with Replacement on Matheric is completely free to use. We believe in accessible education and utility for everyone.

How accurate is Combinations with Replacement?+

We use standard mathematical formulas and high-precision computing algorithms to ensure results for Combinations with Replacement are accurate for academic and professional use.

Can I use Combinations with Replacement on my phone?+

Yes! Combinations with Replacement is fully responsive and optimized for all devices, including smartphones, tablets, and desktops.

Do you save my data?+

No. We prioritize your privacy. All calculations are performed in your browser or temporarily processed, and we do not store your personal input data.

About

Combinations with Replacement (also known as Multiset Coefficients or "Stars and Bars") count the ways to pick $r$ elements from $n$ categories where you can pick the same category multiple times.

Unlike standard combinations, the pool of items doesn't shrink after a selection. This is common in probability problems involving rolling multiple dice or selecting items from a stock that is replenished.

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