Contextualization

Mathematics is one of the oldest academic disciplines, being fundamental to various aspects of our lives. A very important concept within this science is that of permutations, which is the basis of counting and probability. In its simplest scenario, permutations refer to the arrangement of objects in a certain order without repetition. For example, if we have three letters: A, B, and C, we can form six different permutations (ABC, ACB, BAC, BCA, CAB, and CBA).

The study of permutations is fundamental, not only in mathematics, but in many other areas where we need to count or arrange objects in different ways, such as in computer science, statistics, engineering, etc. It is important to emphasize that when it comes to permutations, the order of the elements matters. In other words, the arrangement of the elements affects the final result.

Permutations are a way to explore all the possibilities of a given set, and therefore are frequently used in optimization problems. In the real world, knowledge about permutations is applied in various fields. In the field of computing, for example, permutations help in creating secure passwords and codes. In logistics, it helps optimize delivery routes. In sports, it is used to determine all possible team rankings. It is an important tool that helps explore options and make informed decisions.

For this project, you can base and deepen your knowledge on the subject from the following sources:

• Book: Fundamentals of Elementary Mathematics, Vol.2: Logarithms - Gelson Iezzi. This book is an excellent source for the study of Mathematics and contains a chapter dedicated to combination, permutation, and arrangement.
• Website: Just Mathematics. This website is an incredible tool for learning various mathematical concepts, including permutations.
• Video: Simple Permutation - Education World. This video is very didactic and explanatory, with a simplified example of how permutation works.

Practical Activity

Activity Title: "Permuting Words and Numbers"

Project Objective:

The objective is to develop a project that helps in understanding the concept of permutations and how its application can be carried out through a practical exercise. Through this activity, students will better understand how to solve and elaborate counting and permutation problems, applying the multiplicative principle.

Detailed Project Description:

This project will be carried out by students in groups of 3 to 5 people, with a total execution time of 2 to 4 hours per student. The group must choose words and numbers to explore the possible permutations.

Necessary Materials:

• Paper and pen
• Computer with internet access
• Text editing software (Microsoft Word, Google Docs, etc.)

Detailed Step-by-Step:

1. Each group must choose five words with different quantities of letters and five numbers with different quantities of digits. There should be no repetitions of letters in the chosen words or of digits in the chosen numbers.
2. Using the concept of permutations, each group must calculate how many permutations are possible for each of the words and numbers they chose, remembering that the order of the elements matters in permutations.
3. After calculating the permutations, the group must elaborate a table with the chosen words and numbers and the quantity of possible permutations for each of them.
4. Next, the group must select one of the words and one of the numbers and list all possible permutations for them.
5. To conclude the practical part of the project, the groups must search for real-world applications of the concept of permutations and present at least three examples, explaining how permutation is used in each example.
6. After completing the practical activity, the group must write a report on the project. This report should contain the following items:
   • Introduction: The group must contextualize the project theme, explain its relevance and real-world application, and state the project's objective.
   • Development: The group must explain the theory of permutations, detail the practical activity, indicate the methodology used, and finally present and discuss the results obtained.
   • Conclusion: The group must summarize the main points, explain the learnings obtained, and draw conclusions about the project.
   • References: The group must indicate the sources they relied on to carry out the project, such as books, websites, videos, etc.

This project aims not only at the acquisition of mathematical knowledge about permutations but also at the development of socio-emotional skills, such as time management, communication, problem-solving, creative thinking, and proactivity. Therefore, pay special attention to cooperation and collaboration within your group during the activities. Remember, learning is more effective when it is a collective effort.