Stars and Bars Calculator
Explore combinations with the Stars and Bars method. Distribute items into containers and visualize the possibilities.
Input Parameters
Indistinguishable items to be distributed.
Distinguishable containers to distribute into.
Result
Number of Distributions:
Formula
The number of distributions is calculated using the formula:
Where n is the number of items and k is the number of containers.
Visualization
Understanding Stars and Bars: Items are represented as stars (*) and dividers between containers as bars (|). Below is an example visualization for your inputs:
Note: This visualization shows one possible arrangement of stars and bars. The calculator computes the total number of such arrangements.
Understanding Stars and Bars
The Stars and Bars method is a combinatorial technique used to solve problems where we need to distribute indistinguishable items into distinguishable containers. Imagine you have 'n' identical items (stars) and you want to divide them into 'k' distinct containers. To do this, you need 'k-1' dividers (bars). The problem then becomes arranging these stars and bars in a sequence.
For example, if you want to distribute 5 identical candies (stars) among 3 different children (containers), you would use 3-1=2 bars. A possible arrangement could be * * | * | * *, representing 2 candies for the first child, 1 for the second, and 2 for the third.
The formula for Stars and Bars is given by the binomial coefficient $$ \binom{n+k-1}{n} $$, which calculates the number of ways to choose positions for the 'n' stars (or equivalently 'k-1' bars) out of a total of 'n+k-1' positions. This calculator simplifies this calculation and helps visualize the concept.
Learn more about Stars and Bars on Wikipedia.