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!" CONTENTS CHAPTER 1: INTRODUCTION 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 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 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 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 Scope of analysis and research of topics Permutation with repetition and some problems in the course of discrete mathematics 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: 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 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: Corollary: The number of bijections from a set of elements to a set of elements is II GENERALIZE PERMUTATIONS: 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 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 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 is presented times, other digits are presented once? Solution Each way of making a number has 11 digits so that the number is presented 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 digits, in which is presented times, other digits are presented exactly once? Solution Suppose that the 8-digit number is: + Each way of making a number has digits so that the number appears times and other numbers appear once is a repeated permutation of elements (number appears times; other numbers appear time) (including case a1 = 0) According to the rules of repeated permutation, there are: numbers In the case of a1 = We calculate the number of 7-digit numbers made from 1;2;3;7;9 in which the number is presented 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 appears exactly twice; the number appears twice; the number appears twice and the number appears once are made? Solution Each way of making a number has digits: the number appears exactly twice; the number appears twice; the number appears twice and the number appears once is a repeated permutation of 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 and have digits so that the number appears twice; the number 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 = + The problem becomes the calculation of the number of 6-digit numbers created from the set {1; 3; 6; 9} so that the number appears twice; the number appears twice; and 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 and have digits so that the number appears twice; the number 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 If a8 = The problem becomes the calculation of the number of numbers with digits so that the number appears twice; the number appears twice; the number 6; 7; appeared once According to the rules of repeated permutation, there are: numbers + Similarly, if a8 = 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 appears twice; the number appears twice; the other numbers appear exactly once and if that number is divided by 2, the remainder is Solution Suppose that the satisfied number is: + Since that number leaves as the remainder when we divide it by 2, then that number must be odd ⇒ a7 {5; 7} + Case If a7 = We calculate the number of numbers with digits in which the number appears times; digits 2; 5; 6; appeared exactly once According to the rules of repeated permutation, there are: numbers + Case If a7 = We will calculate the number of numbers with digits so that the number appears times; the number appears twice; The numbers and 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 digits can be made, of which the number is presented 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 = or a8 = + Case If a8 = From now on, the problem becomes finding a number with digits in which the number is presented times; the digits 0; 2; 3; are presented exactly once We have: numbers If the first digit is 0; we’ll find the number of numbers with digits in which the number is presented times and the number 2; 3; appear once We have: numbers ⇒ The number of numbers with digits in which the number is present exactly times; the digits 0; 2; 3;4 are presented exactly once: 840 – 120 = 720 numbers + Case If a8 = At that time; the problem becomes finding a number with digits in which the number is presented times; digits 2; 3; 4; are presented exactly once We have: numbers Therefore, we can make: 720+ 840 = 1560 numbers in total III SOME APPLICATIONS OF PERMUTATION: Protein formation in human body 10 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 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 to 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 11 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 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 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 billion people) However, because the phone numbers also include digits for area codes, they generate fewer numbers than this CHAPTER 3: CONCLUSION 12 The course solved the following problems: Basic knowledge system of permutation Applying repeated permutation in solving some math problems 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 13 [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 Hà Nội [3] Đoàn Quỳnh, Nguyễn Huy Đoan, Nguyễn Xuân Liêm, Nguyễn Khắc Minh, Đặng Hùng Thắng (2007), Đại số giải tích 11 nâng cao, Nhà xuất giáo dục [4] Tạp chí tốn học tuổi trẻ , Nhà xuất Giáo dục [5] Nguyễn Đức Nghĩa, Nguyễn Tô Thành (2009), Giải tích tốn học rời rạc, Nhà xuất Đại học Quốc gia Hà Nội 14 ... analyze, compare, synthesize CHAPTER 2: PERMUTATION WITH REPETITION AND SOME RELATED EXERCISES PART I: PREPARING KNOWLEDGE: PERMUTATIONS I PERMUTATIONS: Definition: A permutation of a set of distinct... 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... is why I choose " Permutation with repetition and some problems " as my research topic Purpose when analyzing the topic Aiming to improve and expand understanding for ourselves and other students,