Possible Selections With Repetition. | Maths Explanation for C# 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 C#, 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 C# 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 C# 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 C#
To generate combinations with repetition in C#, 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 C# combinatorics.
C# 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 C# will work.
Create a new C# class file;
Call it Selection
.
Type out the adjoining C# code for Selection with Repetition.
Why Use This C# Algorithm?
Using a C# 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 C# tutorials
The C# approach shown here can be adapted easily for larger datasets or integrated into more complex applications.
Summary: C# 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 C# recursive algorithms to generate combinations with repetition.
Combinatorics provides the mathematical foundation for solving problems involving group selection. By applying these principles in C# math algorithms, developers can create efficient solutions for tasks such as generating password combinations, simulating probability distributions, or building test cases.
C# Code for Selection With Repetition - Class File
namespace Miscellaneous
{
class Selection
{
public string[] words;
public int r; // min length of word
protected List<string[]> complete_group;
private int i;
public Selection()
{
}
public List<string[]> groupSelection(string[] candidates, int size) {
words = candidates;
r = size;
complete_group = new List<string[]>();
i = 0;
recursiveFillUp(new List<string>());
return complete_group;
}
// pick elements recursively
protected void recursiveFillUp(List<string> temp) {
List<string>[] picked_elements = new List<string>[words.Length];
int j = i;
while (j < words.Length) {
picked_elements[j] = new List<string>();
picked_elements[j].AddRange(temp);
picked_elements[j].Add(words[j]);
// recoil factor
if (i >= words.Length) {
i = j;
}
// satisfied yet?
if (picked_elements[j].Count == r) {
complete_group.Add(picked_elements[j].ToArray());
}
else if (picked_elements[j].Count < r) {
recursiveFillUp(picked_elements[j]);
}
j++;
}
if (picked_elements[--j] != null && picked_elements[j].Count == r) {
i++; // keep recoil factor straightened out
}
}
}
}
C# Code for Selection With Repetition - Main Class
using System.Collections.Generic;
using System.Linq;
namespace Miscellaneous
{
class Program
{
static void Main(string[] args)
{
string[] goods = { "0", "1", "2", "3", "4", "5", "6", "7", "8", "9" };
Selection pick = new Selection();
List<string[]> result = pick.groupSelection(goods, 3);
// print choices and operation
Console.Write("[ ");
foreach (string choice in pick.words)
{
Console.Write(choice + "; ");
}
Console.WriteLine("] selection " + pick.r + ":" + Environment.NewLine);
// print out selections nicely
int i = 0;
foreach (string[] set in result)
{
Console.Write(++i + ": ");
foreach (string member in set)
{
Console.Write(member + "; ");
}
Console.WriteLine();
}
Console.WriteLine(Environment.NewLine + "Number of ways is " + result.Count + ".");
}
}
}