Background

Combinatorics is one of the most versatile and applicable concepts in Mathematics. It is intricately linked with the study of Probability and Statistics but also has extensive uses in other fields like Computer Science, Physics, and even in our day-to-day lives. This project aims to explore and grasp this concept in an engaging manner.

The idea behind combinatorics is about grouping objects together without regard to the order in which they are listed. For example, if we have a group of three people, Alice, Bob and Chris and we want to create pairs out of them, the possible pairings could be Alice and Bob, Alice and Chris, Bob and Chris. Notice that, the pairing Alice and Bob is same as the pairing Bob and Alice because their order doesn’t matter.

In Mathematics, the combination of a set of objects is denoted by C(n, r), where "n" is the total number of items, and "r" is the number of items you wish to select. The formula for finding C(n, r) is n! / r!(n-r)!, where "factorial of a number (n!)," denoted by !, represents the product of all positive integers from 1 to that number.

The core idea behind combinatorics might seem very mathematical, but where combinatorics gets really fascinating is when you realize that it’s at play all around us. If you have ever wondered about the number of ways to create an outfit for yourself using clothes in your closet, or the number of ways a teacher can divide her students into teams, or even how many unique combinations of toppings one can have on their pizza, you’ve encountered combinatorics.

Many real world decision making processes involve some kind of combinatorics analysis. For example, a company that has to select a sub-committee from a larger pool of employees would need to use combinatorics to find out how many possible committees they can form. Further, combinatorics theory is a foundation for Machine Learning and Data Mining, two very active areas of current technological progress.

For a deeper dive into these concepts, we recommend checking out these trusted resources:

1. Khan Academy: Combinatorics - Khan Academy provides free online learning resources, including interactive exercises, videos, and articles on a variety of subjects, including combinatorics.

2. Brilliant.org: Combinations and Permutations - Brilliant offers interactive tutorials, explanations, and problems on a range of math concepts, including combinatorics and permutations.

3. YouTube: Numberphile: Combinations Without Repetition: Socks and Shoes! - A video by Numberphile explaining combinatorics using socks and shoes as an example.

Hands-on Activity

Activity Title: "Combinatorics in Action: The Wardrobe Challenge"

Project Objective

The main goal of this project is to help students understand and apply the concept of combinatorics to practical scenarios. Students will work in teams to explore the number of possible combinations that can be created using a set of items and then analyze how this number changes as the number of items being selected changes.

Detailed Project Description

Students will simulate a scenario where they have a wardrobe with a set quantity of clothing items like pants, shirts, shoes, and accessories. Their goal is to find how many possible outfit combinations they can make. Furthermore, they also have to analyze how this number of possible combinations changes as the number of clothing pieces changes.

Materials Required

1. Paper for calculations
2. Pencil/Pen
3. Calculator
4. Internet for additional research

A Stepwise Guide to Performing the Activity

1. Divide students into groups of 3–5.
2. Each group should create a spreadsheet/document where they list down various types of clothing (e.g. Pants, Shirts, Shoes, Jackets, etc) and the quantity for each type of garment they own.
3. Then, the teams should calculate how many possible outfit combinations can be made if they were to wear ONLY ONE item from each type (i.e., one pants, one shirt, one pair of shoes, etc).
4. Now the groups should repeat the above calculation for various quantities. For example, what if instead of one, they were to wear two different types of shirts? What if they were to change the number of pairs of shoes? And so on.
5. The teams should record all of their calculations and findings in their spreadsheet/documents.
6. Finally, each team should write a short project summary/report where they explain what they did, how they did it, the results they obtained and what they learned through this project.

Project Deliverables

Each group is expected to submit the following as a part of this project:

Written Document

The written report should consist of the following main sections:

1. Introduction: In this section, students should give a context to the topic of combinatorics, their reason to choose a wardrobe as a subject of study, its relevance and real world applications and the objective of their project.

2. Methodology: Here, students should describe the theory of combinatorics used, explain in detail the approach/activity they carried out to complete the task, methodology used, results and observations noted. Calculations and findings should be presented in a well-organized and understandable manner.

3. Conclusion: In this part, teams should summarize their project, highlight their key findings and observations and provide a conclusion. An important part is to give insights into the learning that took place, e.g., about combinatorics and its influence on the possible outfits one can create.

4. References: Lastly, students should give due credit and mention any resources they referred to while working on this project, including books, websites, videos etc.

The report should be submitted as a PDF and must not exceed 10 pages.