Contextualization

The factorial is a fascinating mathematical concept that arises from the multiplicative principle. It plays a central role in various areas of mathematics, from combinatorial analysis to Taylor series in advanced mathematics. The notation "n!" represents the factorial of a non-negative integer "n". The factorial of "n" is the product of all positive integers less than or equal to "n".

The multiplicative principle is a simple rule that determines the number of ways different independent events can occur. If one event can occur in "m" ways, and a second independent event can occur in "n" ways, then the two events together can occur in "m*n" ways. The factorial is a direct application of this principle.

The ability to calculate factorials and understand their properties is fundamental to solving many problems in mathematics and science. For example, in combinatorial analysis, we use factorials to calculate the number of possible permutations or combinations. In probability, we use factorials to calculate the probability of a certain outcome in a random experiment.

The factorial has a wide range of applications, from organizing events to computer programming and modeling real-world situations, such as the probability of winning the lottery or the number of ways to arrange objects. In this project, we will focus on how these concepts can be applied to solving everyday problems.

Understanding the factorial and its applications is very important for mathematics and disciplines that use it, such as physics and engineering. This project will help you understand this concept deeply and apply it effectively.

Practical Activity: "Factoring the World"

Project Objective

The objective of this activity is to apply the concepts of factorial in solving real problems and develop teamwork, critical thinking, and organizational skills.

You are about to embark on a counting and problem-solving mission that will require the power of the factorial. Your group, consisting of 3 to 5 students, will have one week to complete this mission.

Project Description

To carry out this activity, each group will need to follow the detailed steps of the project, research the topic, solve the proposed problems, and prepare a report to present their conclusions.

Materials Needed: Pencil, paper, scientific calculator, and internet access for research.

Project Step by Step

1. Research: Start the project by researching the concept of factorial, how to calculate the factorial of a number, its properties, and its real-life applications.

2. Group Discussion: Based on your research, discuss with your group the meaning of factorial and where it can be applied. This will help everyone understand the topic more clearly.

3. Problem Solving: You will be presented with several hypothetical situations involving the use of the factorial. Your task is to discuss as a group how to approach each problem, solve the problems, and document your solutions.

Hypothetical situations:

• Situation 1: Your group is organizing a school party and needs to determine the number of possible combinations for the order of the songs on the party playlist, which contains 10 different songs. How would you solve it?

• Situation 2: The school is planning a field trip and each student can choose to go to one of 5 different locations. How many different choice combinations are possible for the 25 students in the class?

• Situation 3: Your group needs to create a 4-digit security code that cannot have repeated numbers. How many different codes can be created?

4. Report Preparation: After all situations are solved, the group must prepare a report detailing the entire project process. The structure of the report should be as follows:

   • Introduction: Contextualize the topic, its relevance and real-world application, and the project's objective.
   • Development: Explain the theory of factorial, describe the hypothetical situations, the methodology used to solve them, and present and discuss the results obtained.
   • Conclusion: Conclude by summarizing the main points, explaining the learnings obtained, and drawing conclusions about the project.
   • Bibliography: Indicate the research sources used for the project, such as websites, videos, books, etc.

Once the report is ready, the group must submit it to the teacher for evaluation.

Project Delivery

The project delivery involves presenting the solutions to the proposed problems and the project report. The report should be well-organized, clear, and coherent, demonstrating a clear understanding of the concept of factorial and its applications.

With the completion of this activity, you will have acquired technical skills in solving and elaborating problems involving factorials, as well as socio-emotional skills, such as teamwork, communication, and time management.