Introduction

Mathematics is a discipline that goes beyond numbers, it brings with it a wide variety of concepts and theories that add knowledge and apply to different areas of life. One of these concepts is Permutation with Repetition. This is a concept that belongs to Combinatorial Analysis, an area of mathematics that deals with the counting, arrangement, and permutation of objects within a specific set.

Permutation with repetition is an operation that orders the elements of a set where there are repeated elements. In a permutation without repetition, each element of a set can only be used once. However, in permutation with repetition, each element can be used more than once. To calculate the permutation with repetition, it is generally necessary to divide the factorial of the total number of items by the product of the factorials of the numbers of repeated items.

The study of permutation with repetition is particularly useful in statistics and computer science. It is common in situations where we need to organize or group items in a specific way, especially when some of these items are indistinguishable from each other.

Contextualization

In a world increasingly guided by large volumes of data, permutation with repetition emerges as a crucial tool. For example, in computing, permutation with repetition is used in formulating passwords, where letters, numbers, and symbols can be repeated. This provides a huge number of possible combinations, making breaking these passwords a complex and time-consuming process.

But not only in complex situations, permutation with repetition also applies to everyday situations. For example, choosing a weekly menu in a restaurant, where some dishes can be repeated during the week, is a practical case of permutation with repetition.

To deepen your knowledge on the subject, you can use the following resources:

• Book "Elements of Discrete Mathematics" by Michael O. Albertson and Joan P. Hutchinson. This book addresses several important topics in discrete mathematics, including Combinatorial Analysis.

• Website "Brasil Escola" (https://brasilescola.uol.com.br/matematica/permutacao-elementos-repetidos.htm). It is an online resource that simplifies complex topics in easy-to-understand language. The link leads to a detailed explanation of Permutation with Repeated Elements.

• Website "Só Matemática" (https://www.somatematica.com.br/emedio/analise2.php). Here, you will find complete didactic material and exercises on Combinatorial Analysis.

Thinking about how these concepts are widely applicable, let's now embark on a journey through permutation with repetition.

Practical Activity

"Unveiling the Code: Exploring Permutations with Repetition"

Project Objective

This project aims to provide a deep understanding of the concept of Permutation with Repetition. For this, we will create a puzzle game involving the discovery of a secret code!

Students will work in groups of 3 to 5 people to design and solve puzzles based on this concept. This project will stimulate logical thinking, collaboration, and time management skills.

Detailed Project Description

1. Concept Study and Planning

Each group should start the project by studying the concept of Permutation with Repetition through the resources suggested in the introduction, as well as others that the group finds interesting. Once the group feels comfortable with the concept, they should start planning their puzzle. For the code, you can use any combination of numbers, letters, or symbols with repetitions.

2. Puzzle Design

Each group will design a puzzle that involves discovering a secret code based on the concept of Permutation with Repetition. Be creative! The difficulty of the puzzle should be such that it is challenging but not impossible to solve. Remember to create a script with clues to solve the puzzle.

3. Puzzle Exchange

Each group will exchange puzzles with another group in the class. Each group will have to solve the puzzle they receive, recording the resolution process.

4. Presentation of the Resolution Process

Each group, in turn, will present to the class the resolution of the puzzle, explaining the logic used and the application of the concept of Permutation with Repetition in the process.

Necessary Materials

1. Paper and pen for sketches and notes.
2. Computer with internet access for research.
3. Presentation software for the elaboration of the final presentation (Example: Google Slides, PowerPoint).

Activity Step by Step

1. Study the concept of Permutation with Repetition.
2. Plan the puzzle based on the studied concept.
3. Design the puzzle with your secret code.
4. Exchange puzzles with another group.
5. Solve the received puzzle.
6. Prepare a presentation explaining the resolution process and the application of the concept.
7. Present to the class.

Project Delivery

1. Written Report: The report must contain the sections of Introduction, Development, Conclusions, and Bibliography. In the introduction, the group must contextualize the project, explain the concept of Permutation with Repetition, and discuss its relevance. In the development, the group must describe the methodology used to design the puzzle, the resolution process, and the application of the concept of permutation with repetition in this process. In the conclusion, the group must review key points and discuss the learnings obtained.

2. Presentation: The presentation should include an explanation of the puzzle created by the group, the solution to the received puzzle, and a discussion on the application of the concept of Permutation with Repetition in solving the puzzle.