PERMUTATION WITH REPETITION AND SOME RELATED EXERCISE

In the process of learning and training, the right acquisition, sufficiently according to the knowledge and skills standards of the training program is the central task of each student in general. However, besides, a task is equally important and necessary for students that is to selfnurture, deepen knowledge, expand and improve the knowledge that has been guided by lecturers. This not only helps students master basic skills but also practice the habit of thinking, learning, reasoning, solving a problem, a difficult problem in a strict and logical way. Thereby forging students’ intelligence, creativity, interest in discrete mathematics.Being aware of the importance of the problem after carefully studying some of the relevant documents, I boldly came up with a system of knowledge about Permutation with repetition and some problems. I hope that this topic will more or less help other students in fostering their own knowledge of the mean value theorem, which is why I choose Permutation with repetition and some problems as my research topic.

THAI NGUYEN UNIVERSITY OF EDUCATION

MATHEMATICS

- -PROBLEM SEMINAR

PROBLEM: PERMUTATION WITH REPETITION

AND SOME RELATED EXERCISE

Supervisors: Ph.D TRAN NGUYEN AN Author: NGUYEN VAN TRANG

GIAP THI THUC TRINH

NGUYEN NHU QUYNH

MAC TIEN DUNG

Class: English of mathematics (NO1)

June, 2022

THANK YOU

"First of all, we would like to express our appreciation to Thai Nguyen University of Education for introducing discrete mathematics into the curriculum In particular, we would like to express my deep gratitude to my lecturer – Mr Tran Nguyen An for his support teaching and imparting valuable knowledge to us during our last study period

as well as thanking the Library for lending us books and materials so that we could study and work effectively

In your discrete math class, we have gained more useful knowledge, effective and serious study spirit These will definitely be valuable knowledge, a luggage for us to step up later Discrete mathematics is an interesting, extremely useful, and highly practical subject Guaranteed to provide enough knowledge, tied to students' real needs However, due to limited knowledge and accessibility Although we have tried our best, it is certain that the essay cannot avoid errors and inaccuracies in many places We hope you will review and comment to make our essay more complete

We want to sincerely thank you!"

CHAPTER 1: INTRODUCTION

1. Reasons for choosing a topic

Although it has been more than two thousand years, mathematics has proven itself

as a peak of human intelligence, intruding into most sciences and the foundation of many important scientific theories Today with the age of advanced industry and the rapid development of information technology, the role of mathematics becomes more important and necessary than ever

In the process of learning and training, the right acquisition, sufficiently according to the knowledge and skills standards of the training program is the central task of each student in general However, besides, a task is equally important and necessary for students that is to self-nurture, deepen knowledge, expand and improve the knowledge that has been guided by lecturers This not only helps students master basic skills but also practice the habit of thinking, learning, reasoning, solving a

problem, a difficult problem in a strict and logical way Thereby forging students’ intelligence, creativity, interest in discrete mathematics

Through the time of participating in the class, I noticed that the knowledge of the

"Permutation" is the central, basic knowledge, often used in the curriculum Therefore, I realized that our students need to master this knowledge In particular, it

is necessary to have a clear and complete view of the " Permutation with repetition and some problems " to apply in solving some related problems as well as in practice

Being aware of the importance of the problem after carefully studying some of the relevant documents, I boldly came up with a system of knowledge about Permutation with repetition and some problems I hope that this topic will more or less help other students in fostering their own knowledge of the mean value theorem, which is why I choose " Permutation with repetition and some problems " as my research topic

2. Purpose when analyzing the topic

Aiming to improve and expand understanding for ourselves and other students, especially self-fostering, gives us a fuller view of Permutation with repetition and some problems Thereby helping us to improve the methods of solving problems related to solve some related problems and training creative thinking for ourselves

3. Tasks when analyzing the topic

To give the most general view of the "Permutation with repetition" from the definition, the theorem, the corollary and how to prove the iterative permutation formula, then apply it to solve related problems

4. Subject of study

The object to be analyzed is the " Permutation with repetition." Specifically, the definition, the theorem, the corollary and how to prove the iterative permutation formula, then apply it to solve related problems

5. Scope of analysis and research of topics

Permutation with repetition and some problems in the course of discrete mathematics

6. Methods of analysis and research of topics

Read documents, analyze, compare, synthesize

CHAPTER 2: PERMUTATION WITH REPETITION AND SOME RELATED EXERCISES

PART I: PREPARING KNOWLEDGE: PERMUTATIONS

I PERMUTATIONS:

1 Definition:

A permutation of a set of distinct objects is an ordered arrangement of these objects

An ordered arrangement of r elements (or an ordered r-arrangement) of a set is called

an r-permutation The number of r-permutations of a set with n elements is denoted

by or If , we denoted by

2 Theorem:

- If is a positive integer and r is an integer with , then there are

r-permutations of a set with n distinct elements.

- The number of permutations of n elements:

3 Corollary:

The number of bijections from a set of elements to a set of elements is

II GENERALIZE PERMUTATIONS:

1 Definition:

An ordered r-arrangement with repetition of the elements of the set with n elements is also called an r-permutations of a set with n elements when repetition is allowed The number of r-permutations of a set with n elements when repetition is allowed is

denoted by

2 Theorem:

The number of r-permutations of a set of n objects with repetition allowed is

PART II: PERMUTATION WITH REPETITION

I DEFINITION OF PERMUTATION WITH REPETITION:

For n elements, including n1 element x1, n2 elements x2, , nk element xk

(n1+n2+ +nk=n) Each way of arranging n that element into n position is called

a repeated permutation of the given element n

The number of all repeated permutations of n elements above is:

Proof.

To determine the number of permutations, we see that there are ways to keep n elements of type 1, remaining spaces left

Then there are ways to put elements of type 2 into the permutation, remaining spaces

Continue to put elements of type 3, type 4, …, type into spaces in the permutation Hence, we have ways to put elements of type k into the permutation According to the multiplication rule, all possible permutations are:

II EXERCISES:

Example 1: From set X = {1;2;3;4;5;6;7;8} How many natural numbers have 11

digits so that the number 1 is presented 4 times, other digits are presented once?

Solution.

Each way of making a number has 11 digits so that the number 1 is presented 4 times, other digits are presented once is a repeated permutation of 11 elements

According to the rules of repeated permutation, there are:

Example 2: With the digits 0; 1; 2; 3; 7; 9, how many numbers can be made of 8

digits, in which 1 is presented 3 times, other digits are presented exactly once?

Solution.

Suppose that the 8-digit number is:

+ Each way of making a number has 8 digits so that the number 1 appears 3 times and other numbers appear once is a repeated permutation of 8 elements (number

1 appears 3 times; other numbers appear 1 time) (including case a1 = 0)

According to the rules of repeated permutation, there are:

numbers

In the case of a1 = 0

We calculate the number of 7-digit numbers made from 1;2;3;7;9 in which the number 1 is presented 3 times, and other numbers are presented exactly once

According to the rules of repeating permutation, there are:

numbers

Therefore, from the given digits, we can make:

6720 - 840 = 5880 numbers

Example 3: From the numbers of set A= {2; 4; 6; 8}, how many natural

numbers of seven digits, in which the number 2 appears exactly twice; the number 4 appears twice; the number 6 appears twice and the number 8 appears once are made?

Solution.

Each way of making a number has 7 digits: the number 2 appears exactly twice; the number 4 appears twice; the number 6 appears twice and the number 8 appears once is a repeated permutation of 7 elements

According to the rules of repeated permutation, we can make:

numbers

Example 4: Given set A= {1; 3; 5; 6; 9} From A, how many numbers that are

divisible by 5 and have 7 digits so that the number 1 appears twice; the number 6 appears twice; other numbers appear exactly once are made?

Solution.

Suppose that the satisfied number is:

Since this number is divided by 5, then a7 = 5

+ The problem becomes the calculation of the number of 6-digit numbers created from the set {1; 3; 6; 9} so that the number 1 appears twice; the number 6 appears twice; 3 and 9 only appear once

According to the rules of repeated permutation, there are:

numbers that satisfy the problem

Example 5: From set X= {1; 2; 4; 6; 7; 9} From set X, how many numbers that

are NOT divisible by 2 and have 8 digits so that the number 4 appears twice; the number 2 appears twice; other digits appear exactly once are made?

Solution.

Suppose that the satisfied number is:

Since this number is not divisible by 2, a8 {1; 7; 9}

+ Case 1 If a8 = 1 The problem becomes the calculation of the number of numbers with 7 digits so that the number 4 appears twice; the number 2 appears twice; the number 6; 7; 9 appeared once

According to the rules of repeated permutation, there are:

numbers

+ Similarly, if a8 = 7 or 9, we also have 1260 satisfied numbers

Therefore, from the set X, we can make:

1260+ 1260+ 1260 = 3780 numbers

Example 6: For set A= {2; 4; 5; 6; 7} How many 7-digit numbers can be made

so that the number 4 appears twice; the number 5 appears twice; the other numbers appear exactly once and if that number is divided by 2, the remainder is 1

Solution.

Suppose that the satisfied number is:

+ Since that number leaves 1 as the remainder when we divide it by 2, then that number must be odd

⇒ a7 {5; 7}

+ Case 1 If a7 = 5.

We calculate the number of numbers with 6 digits in which the number 4 appears

2 times; digits 2; 5; 6; 7 appeared exactly once

According to the rules of repeated permutation, there are:

numbers

+ Case 2 If a7 = 7.

We will calculate the number of numbers with 6 digits so that the number 4 appears 2 times; the number 5 appears twice; The numbers 4 and 6 appear exactly once

According to the rules of repeated permutation, there are:

numbers

Combining the two cases, we can make:

360 + 180 = 540 numbers

Example 7: With the digits {0; 1; 2; 3; 4; 5} how many numbers of 8 digits can

be made, of which the number 1 is presented 3 times, other digits are presented exactly once and that number is divisible by 5?

Solution.

Suppose that the 8-digit number is:

Since the number is divisible by 5, then a8 = 0 or a8 = 5

+ Case 1 If a8 = 5.

From now on, the problem becomes finding a number with 7 digits in which the number 1 is presented 3 times; the digits 0; 2; 3; 4 are presented exactly once

We have:

numbers

If the first digit is 0; we’ll find the number of numbers with 6 digits in which the number 1 is presented 3 times and the number 2; 3; 4 appear once

We have:

numbers

⇒ The number of numbers with 7 digits in which the number 1 is present exactly

3 times; the digits 0; 2; 3;4 are presented exactly once:

840 – 120 = 720 numbers

+ Case 2 If a8 = 0.

At that time; the problem becomes finding a number with 7 digits in which the number 1 is presented 3 times; digits 2; 3; 4; 5 are presented exactly once

We have:

numbers

Therefore, we can make:

720+ 840 = 1560 numbers in total

III SOME APPLICATIONS OF PERMUTATION:

1 Protein formation in human body.

Proteins are formed from string-like structures that contain an arrangement of amino acids

The sequence of amino acids is important because this will determine whether the protein will function or not

Defective or missing proteins can cause serious diseases in people such as sickle cell anemia

For example, insulin is a protein found in humans Its role is to control the amount

of sugar around the body so that it is neither too high nor too low

Insulin is made up of 51 amino acids arranged in a specific sequence or permutation

Any rearrangement that deviates from this normal sequence makes this protein dysfunctional and causes diseases such as diabetes

The body has a mechanism to ensure that this sequence is followed and the correct protein is formed

2 Working out a way to win a lottery.

If you play a lottery game, you may want to know what your odds are for winning the prize

For example, if the lottery rules say you can win if you pick four digits that match (e.g., 1111, 9999, or 5555), you can work out your odds for winning by using a permutation calculation

Each slot (digit position) can be occupied in 10 different ways (since we have 0 to 9 digits)

So, we can calculate the total numbers of possible 4-digit numbers as 10 x 10 x 10

x 10 = 10,000 different numbers

From there, we count the total of winning numbers (0000, 1111,2222, 3333…… 9999) which are 10

So, there are 10 numbers that can win and the total numbers you can draw are 10.000

Therefore, you have a probability of picking the right sequence of 10/10,000

=0.001

In percentage terms, you have a 0.1% (0.001 x 100) chance of winning

3 Finding out the number of available phone numbers.

No two phone numbers are supposed to be alike

Using permutations, a phone company can determine the number of unique telephone numbers it can issue based on the number format it wants to use

If the phone numbers should all consist of 10 digits and can use 0 to 9, an example

of one such number could be: 5653278065

We can work out the total numbers that can be available in this way: each slot or digit position can be occupied by any of the 10 digits (0 to 9)

Since repetition is allowed, this works out to:

numbers

These are a lot of numbers that easily outnumber the population of the world (5 or 6 billion people)

However, because the phone numbers also include digits for area codes,

they generate fewer numbers than this

CHAPTER 3: CONCLUSION

The course solved the following problems:

1. Basic knowledge system of permutation

2. Applying repeated permutation in solving some math problems

3. Some applications of permutations

In the process of studying the document due to the limited time, after completing it,

we will continue to study more deeply about permutations with repetition and its applications We look forward to receiving your comments to improve our essay We sincerely thank you

CHAPTER 4: REFERENCES

[1] Donald Knuth, The Art of Computer Programming, Volume 3: Sorting and

Searching, Third Edition Addison-Wesley, 1997.

[2] Lê Hồng Đức, Lê Bích Ngọc, Lê Hữu Trí (2003), Phương pháp giải toán tổ hợp,

Nhà xuất bản Hà Nội

[3] Đoàn Quỳnh, Nguyễn Huy Đoan, Nguyễn Xuân Liêm, Nguyễn Khắc Minh, Đặng

Hùng Thắng (2007), Đại số và giải tích 11 nâng cao, Nhà xuất bản giáo dục.

[4] Tạp chí toán học tuổi trẻ , Nhà xuất bản Giáo dục.

[5] Nguyễn Đức Nghĩa, Nguyễn Tô Thành (2009), Giải tích toán học rời rạc, Nhà

xuất bản Đại học Quốc gia Hà Nội