Possible Selections With Repetition. | Maths Explanation for Visual Basic Kids
Selection with repetition is a core concept in combinatorial mathematics and appears frequently in probability theory, algorithm design, and computational problem-solving. In this tutorial, we demonstrate how to implement selection with repetition in Visual Basic, allowing elements to be chosen multiple times while generating all valid combinations.
This page focuses on the algorithmic approach, explains the underlying logic, and provides a clear Visual Basic implementation suitable for tertiary-level study.
What Is Selection With Repetition?
In combinatorics, selection with repetition (also known as *combinations with repetition*) refers to the process of selecting items from a set where each item may be chosen more than once and where order does not matter.
For example, selecting 3 items from {A, B, C} with repetition allowed produces combinations such as:
- A, A, A
- A, A, B
- A, B, B
- B, C, C
This concept differs from permutations and combinations without repetition, where either order matters or repeated elements are disallowed.
Imagine being given the opportunity to pick six (6) exercise books
from a book-shop having 10 different brands of exercise books -
Ben 10, Chelsea F.C., Superman, Tottenham F.C.,
Indomitables, Manchester City F.C., Spider Man,
Power Rangers, Liverpool F.C. and Bat Man exercise books.
If you happen to be a big Power Rangers fan, nothing stops you from
selecting all 6 exercise books to be Power Rangers exercise books.
But you can as well decide to pick only 3 Power Rangers exercise books
and make up for the other 3 with any other brands of exercise book.
Whether you are calculating possible outcomes for a game, managing inventory combinations, or building a Visual Basic math algorithm, understanding how to handle multisets is essential.
The nCr with Repetition Formula
To find the total number of ways to select items when repetition is allowed, we use a variation of the standard binomial coefficient. The formula for combinations with repetition is:
Where:
- n = The number of distinct types of items to choose from.
- r = The number of items being selected.
This is often visualized using the stars and bars method, where "stars" represent the items and "bars" represent the dividers between different categories.
Selection With vs Without Repetition
| Feature | With Repetition | Without Repetition |
|---|---|---|
| Repeated elements allowed | Yes | No |
| Order matters | No | No |
| Common use cases | Probability, modelling | Sampling, grouping |
| Algorithm style | Recursive / Iterative | Recursive / Iterative |
Understanding this distinction is essential when solving combinatorial selection problems in programming.
Selection With Repetition in Visual Basic
To generate combinations with repetition in Visual Basic, we typically use a recursive algorithm. Recursion allows us to build combinations incrementally while controlling the index from which elements may be selected, ensuring that duplicates and invalid orderings are avoided.
This approach is efficient, readable, and well suited for teaching Visual Basic combinatorics.
Visual Basic Selection With Repetition Algorithm
The algorithm for Selection with Repetition will be a lot similar to that of combination.
The selection with repetition algorithm follows these steps:
- Start with an empty selection array.
- At each recursive call, allow selection from the current index onward.
- Add the selected element to the current combination.
- Repeat until the desired selection length is reached.
- Store or output the completed combination.
By restricting future selections to the current index or higher, the algorithm ensures that combinations are generated without duplication while still allowing repetition.
This is how our Repetitive Selection code in VB.Net will work.
Create a new VB.Net class file; Call it Selection
.
Optionally, Create a new VB.Net module file; Call it Miscellaneous_Selection
.
Type out the adjoining VB.Net algorithm for Selection with Repetition.
Why Use This Visual Basic Algorithm?
Using a Visual Basic combinations algorithm is vital for web-based tools that require real-time probability calculations. Common use cases include:
- Multiset Generation: Listing all unique ways to group items.
- Sampling with Replacement: Statistical modeling where items are returned to the pool after selection.
- Resource Allocation: Distributing r identical units into n distinct bins.
Applications of Selection With Repetition
Selection with repetition is commonly used in:
- Generating test cases for software development
- Modeling probability and statistics problems
- Creating secure password or key combinations
- Resource allocation problems
- Algorithm design exercises
- Exploring mathematical concepts like combinatorics in Visual Basic tutorials
The Visual Basic approach shown here can be adapted easily for larger datasets or integrated into more complex applications.
Summary: Visual Basic Selection with Repetition Algorithm
Selection with repetition is a fundamental concept in combinatorics that allows us to choose items from a set where each element can be selected more than once. In programming, this is often implemented using Visual Basic recursive algorithms to generate combinations with repetition.
Combinatorics provides the mathematical foundation for solving problems involving group selection. By applying these principles in Visual Basic math algorithms, developers can create efficient solutions for tasks such as generating password combinations, simulating probability distributions, or building test cases.
VB.Net Code for Selection With Repetition - Class File
Public words As String()
Public r As Integer ' min length Of word
Protected complete_group As List(Of String())
Private i As Integer
Public Function groupSelection(candidates As String(), size As Integer) As List(Of String())
words = candidates
r = size
complete_group = New List(Of String())
i = 0
recursiveFillUp(New List(Of String))
Return complete_group
End Function
' pick elements recursively
Protected Sub recursiveFillUp(temp As List(Of String))
Dim picked_elements(words.Length) As List(Of String)
Dim j As Integer = i
Do While j < words.Length
picked_elements(j) = New List(Of String)
picked_elements(j).AddRange(temp)
picked_elements(j).Add(words(j))
' recoil factor
If i >= words.Length Then
i = j
End If
' satisfied yet?
If picked_elements(j).Count = r Then
complete_group.Add(picked_elements(j).ToArray())
ElseIf picked_elements(j).Count < r Then
recursiveFillUp(picked_elements(j))
End If
j += 1
Loop
j -= 1
If picked_elements(j).Equals(Nothing) = False And picked_elements(j).Count = r Then
i += 1 ' keep recoil factor straightened out
End If
End Sub
End Class
VB.Net Code for Selection With Repetition - Main Module
Sub Main()
Dim goods As String() = {"0", "1", "2", "3", "4", "5", "6", "7", "8", "9"}
Dim pick As New Selection
Dim result As List(Of String()) = pick.groupSelection(goods, 2)
' print choices and operation
Console.Write("[ ")
For Each choice As String In pick.words
Console.Write(choice & "; ")
Next
Console.WriteLine("] selection " & pick.r & ":" & Environment.NewLine)
' print out selections nicely
Dim i As Integer = 0
For Each group As String() In result
i += 1
Console.Write(i & ": ")
For Each member As String In group
Console.Write(member & "; ")
Next
Console.WriteLine()
Next
Console.WriteLine(Environment.NewLine & "Number of ways is " & result.Count & ".")
End Sub
End Module