Stars and Bars Use Cases

1. Distributing Objects into Groups:

We can use Stars and Bars Approach to find the number of ways to distribute N objects among K groups for both the cases, when a group can have zero objects as well as when a group should have at least one object. The approach as well as the implementation has been discussed above.

2. Finding the number of Solutions of Equations:

The Stars and Bars approach can also be used to find the number of solutions to the equations like x1 + x2 + x3 …. = S, such that xi >= 1. For example: x1 + x2 + x3 = 10, where x1, x2, and x3 are non-negative integers.

Here, the stars would represent units of 1, and the bars would separate the different variables. According to the Stars and Bars theorem, there are C(10+3-1, 3-1) = C(12, 2) = 66 solutions to this equation.

Similarly, if x1, x2 and x3 are positive numbers then according to the Stars and Bars theorem, there are C(10 – 1, 3 – 1) = C(9, 2) = 36 solutions to this equation.

3. Finding the number of Solutions of Equations with a lower bound restrictions on Solutions:

The Stars and Bars approach can also be used to find the number of solutions to the equations like x1 + x2 + x3 …. = S, when we are given lower bounds that x1 >= y1, x2 >= y2 and x3 >= y3… and so on.

So, if we substitute x1 with x1′, x2 with x2′ and xi with xi’ such that xi’ = (xi – yi). So, the new equation would be:

(x1′ + y1) + (x2′ + y2) + (x3′ + y3) …. = S, which can be further simplified to

x1′ + x2′ + x3′ …. = S – y1 – y2 – y2…, such that x1′, x2′, x3′ …. are >= 0.

Now, we can again use Stars and Bars approach as we used in the above use case to get the number of solutions.

The Stars and Bars (also known as Balls and Urns) technique is a popular method used in Combinatorics, the study of counting and arrangement. It’s a graphical way to solve certain types of problems and is particularly useful when dealing with problems related to the distribution of identical objects into distinct groups. This article will introduce the concept of Stars and Bars algorithm in depth, including some examples and a detailed explanation of how it is implemented. In addition, we are going to show you some of the use cases it has with real-life applications.