Contextualization

The study of permutations with repetition falls within combinatorial analysis, a branch of mathematics that, among other things, deals with calculating the number of possibilities for an event to occur. When dealing with permutations, we are basically talking about arrangements, that is, different ways of ordering a set of elements. The highlight here is that we are dealing with this ordering when some of these elements are repeated.

Permutation with repetition appears in many scenarios of our daily lives. For example, in the definition of alphanumeric passwords, where it is allowed to use the same letter or number more than once. In this case, to ensure the security of the password, we want there to be many possible permutations, which is allowed by the repetition of the characters. Another classic example is genetics, where the combination of genes (with repetitions) defines the characteristics of living beings.

Introduction

Permutation with repetition is a concept of combinatorial theory that allows us to calculate the total number of arrangements or sequences that can be made with a set of elements, where some of these elements are repeated. It is an intriguing and challenging topic due to the complexity that the repetition of elements adds to the calculation of permutations.

In general terms, a permutation of a set of elements is simply a reorganization of these elements in a specific order. However, when we come across the repetition of elements, the number of possible permutations changes. To understand this, consider the word "BANANA". If all the letters were distinct, we would have a total of 6! (that is, 6 factorial) possible permutations. However, since we have repeated letters, the number of permutations will be smaller.

Hands-on Activity

Title: Unveiling Permutations with Repetition: From Theory to Practice

Project Objective

The objective of this project is to understand, in a practical and engaging way, the concept and application of permutation with repetition. Students will create a password game based on the idea of permutation with repetition, putting into practice their mathematical skills and teamwork skills.

Detailed Project Description

In this activity, students will develop and play a password guessing game. The game will be built from a set of pieces (it can be anything at hand such as letters, numbers, colored chips), where some pieces are repeated. The goal of the game is to guess the password assembled by the opponent, considering the rules of permutation with repetition.

Required Materials

• Set of pieces (letters, numbers, color games, etc.)
• Paper, pencil and eraser for notes
• Calculator

Detailed Step-by-Step to Carry out the Activity

1. Group Formation: Students should be divided into groups of 3 to 5 members. The choice of groups can be made by the teacher or by the students, as they prefer.

2. Previous Study: Before starting the practical activity, students should study the concept of permutation with repetition, using the suggested resources or other additional materials they find. It is important that they understand the concept well before going into practice.

3. Game Creation: The groups must now create a game based on a password that can have repetitions of characters. They must decide how many and which elements will make up their password (for example, a password with 5 numeric digits from 0 to 9 can repeat numbers).

4. Playing: Once the game is ready, the group members play with each other, trying to guess each other's password. They should record each attempt and the total number of attempts until the password is correct.

5. Mathematical Analysis: After the game is over, students should analyze the results in light of the theory of permutation with repetition. How many permutations were possible? How many attempts were needed on average to get the password right? Is the result in accordance with the theory learned?

6. Final Report: Based on the experiences of the game and the analyzes made, the groups must prepare a final report. The report should include sections for introduction, development, conclusion, and bibliography.

Project Deliverables

The main product of this project is the final report. It should describe the process of creating and developing the game, as well as analyze the results obtained in light of the theory of permutation with repetition. The report should contain:

• Introduction: Contextualize the theme of permutation with repetition in mathematics and in real-world applications. The objective of the report and this project should also be stated.
• Development: A detailed description of the game, explaining how it was conceived, developed and what the results were. Here, students should explain the theory of permutation with repetition, connect with the practical activity carried out, the methodology used and present/discuss the results obtained.
• Conclusion: Students should summarize what they learned from the project and how it deepened their understanding of permutation with repetition. What did this project teach them about math and teamwork?
• Bibliography: References to the materials used to study the topic and develop the project.

The duration of the project is one month, and the estimated workload for each student is 5 to 10 hours.