INTRODUCTION

Relevance of the Topic

Combinatorial Analysis is a fundamental pillar within the field of Mathematics. This study allows us to understand and quantify the different arrangements, combinations, and possibilities that can be formed from a set of elements, becoming crucial in a multitude of scenarios, from exact sciences to human sciences.

Among the fundamentals of Combinatorial Analysis, Permutation with Repetition stands out as a powerful and versatile concept that, once mastered, can be applied in a variety of practical situations. Understanding it will prepare you for the study and application of more advanced topics in Mathematics.

Contextualization

In the Mathematics curriculum of the 2nd year of High School, Combinatorial Analysis is deepened after the study of Probability, since its techniques provide the basis for calculating possibilities in different scenarios.

Permutation with Repetition emerges as a natural unfolding of Simple Permutation, previously explored in the classroom. Here, we are dealing with elements that repeat, which considerably broadens the range of scenarios where this technique can be applied.

At this point in the subject, students are expected to already be familiar with factorial notation and the basic applications of permutation and combination, making them ready for this new approach. Mastery of Permutation with Repetition will allow them to solve more complex questions of Combinatorial Analysis and will be present in mathematical challenges and practical applications in various areas of knowledge.

THEORETICAL DEVELOPMENT

Components

• Permutation with Repetition of identical elements: Crucial section of the study of Permutation with Repetition where we explore the possibility of permuting a set in which some or all elements are repeated. Here, order and repetition matter.
• Factorial and its developments: A deep understanding of factorial notation is fundamental for the manipulation and resolution of permutation with repetition problems. Knowing the properties that govern this mathematical operation is essential to enhance the resolution of such problems.
• Identification of the elements of a set: The ability to identify how many times each element of a set repeats is of utmost importance in permutation with repetition. This step ensures that there is no overestimation or underestimation of the total number of possible arrangements.

Key Terms

• Permutation with Repetition: Allows calculating the number of possible arrangements of a set where some or all elements are repeated.
• Factorial: Product of all positive integers from 1 to n, denoted by n!. Plays an important role in the calculation of permutations and combinations.
• Set: Collection of distinct elements without specific order. The values in the set are considered unique, even if they are repeated.

Examples and Cases

• Example 1 - Permutation with Repetition of Letters: Given the set of letters {A, B, C, C, D}, how many 3-letter words can we form? The application of the permutation with repetition formula is fundamental in this case.
• Example 2 - Permutation with Repetition of Numbers: In a decoration event, we need to place 5 identical balls on a Christmas tree with 3 branches. How many possibilities do we have? Although all the balls are the same, the position of each one is considered, characterizing a problem of permutation with repetition.
• Example 3 - Identification of Repeated Elements: In the set of numbers {1, 1, 2, 3, 4} how many permutations of 3 distinct numbers can we form? Here it is necessary to correctly identify the number of times each number repeats, as the permutation with repetition formula uses this information.

DETAILED SUMMARY

Relevant Points

• Definition of Permutation with Repetition: It is the technique we use to calculate how many arrangements can be made from a set where some or all elements are repeated.
• Factorial and its relation to Permutation with Repetition: The factorial is the basis for solving permutation with repetition problems. Understanding and correctly applying its concept and properties is crucial.
• Identification of Repeated Elements: Correct and precise identification of which elements repeat and how many times will help not to overestimate or underestimate the total number of permutations possible.

Conclusions

• Permutation with Repetition is a key concept within the study of Combinatorial Analysis, which allows for efficient counting of possible arrangements when dealing with elements that repeat in a set.
• The factorial notation and its deep understanding are fundamental for the manipulation and resolution of permutation with repetition problems.
• Correct identification of repeated elements is a crucial step in solving permutation with repetition problems.

Suggested Exercises

1. Exercise 1 - Word Counting: In a word formed by the letters of the word "ABACAXI", how many permutations are possible considering that all letters must be used and the letters "A" and "X" cannot be repeated?
2. Exercise 2 - Balls in a Bag: How many different arrangements can we have if we want to take 3 balls from a bag containing 4 red balls, 3 blue balls, and 2 green balls?
3. Exercise 3 - Sequences of Numbers: Given the set {2, 2, 5, 5, 6, 6}, how many 4-digit distinct numbers can we form?