Combinations Calculator

🔄 Multiset Combinations

Calculate combinations with replacement - items can be chosen repeatedly

Types of Items (n)

Items to Choose (r)

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

Example: Ice Cream Scoops

If there are 5 flavors and you want 3 scoops, you can choose the same flavor multiple times. This is different from regular combinations where each item can only be chosen once.

📊 Results

Combination with replacement calculation results

Enter values to calculate combinations with replacement

📈 Growth Comparison

Combinations with vs without replacement

🌐 3D Visualization

Interactive 3D representation of multiset combinations

Use mouse to rotate and zoom

Combinations with Replacement Guide

Understanding multiset combinations and their applications

Combinations with replacement allow selecting the same item multiple times. This is also called "combinations with repetition" or "multiset combinations." The formula C(n+r-1, r) accounts for the fact that we can choose each type of item any number of times up to r total selections.

Replacement Allowed

Items can be selected multiple times, unlike regular combinations

Stars and Bars

Mathematical technique using dividers to count multiset combinations

Real Applications

Used in probability, statistics, and combinatorial optimization

Growth Pattern

Grows faster than regular combinations for the same parameters

Common applications include: selecting flavors for ice cream scoops, choosing lottery numbers with repetition allowed, distributing identical objects into distinct boxes, and probability calculations where events can repeat.

Combination with Replacement Calculator