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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.

GROUP Tìm kiếm Google Xem Trang tìm WELCOME TO GROUP MEMBERS OF THE GROUP Nguyễn Như Quỳnh Giáp T.Thục Trinh Nguyễn Vân Trang Start the description Permutation with repetition Discrete mathematics Mạc Tiến Dũng Thành viên Trong Members of thenhóm group LECTURERS : TRAN NGUYEN AN Permutation with repetition PERMUTATION WITH REPETITION AND SOME RELATED LESSONS Start the description Discrete mathematics Thành viên Trong Members of thenhóm group Discrete mathematics CONTENT 1: PREPARING KNOWLEDGE PERMUTATIONS CONTENT 2: PERMUTATION WITH REPETITION Tên Tên : Mã Sinh Viên CONTENT Mã3 Sinh Viên EXERCISES CONTENT 4: SOME APPLICATIONS OF PERMUTATION Tên Tên Mã Sinh Viên Mã Sinh Viên Start the description PERMUTATION PERMUTATION PERMUTATION WITH REPETITION AND SOME RELATED LESSONS PERMUTATION WITH REPETITION AND SOME RELATED LESSONS Discrete mathematics Thành viên Trong Members of thenhóm group CONTENT 1: PREPARING KNOWLEDGE PERMUTATIONS CONTENT 2: PERMUTATION WITH REPETITION Tên Tên : Mã Sinh Viên CONTENT Mã3 Sinh Viên EXERCISES CONTENT 4: SOME APPLICATIONS OF PERMUTATION Tên Tên Mã Sinh Viên Mã Sinh Viên Start the description PERMUTATION PERMUTATION PERMUTATION WITH REPETITION AND SOME RELATED LESSONS PERMUTATION WITH REPETITION AND SOME RELATED LESSONS Discrete mathematics Permutation with repetition Thành viên Trong Members of thenhóm group Content Content START THE DESCRIPTION Chủ Tênđề thuyết trình Tên Tên Tên Mã Sinh Viên Content Mã Sinh Viên Content GiáoSinh viên Viên môn: Cô Nguyễn A Viên Mã MãVăn Sinh Members of the group Discrete mathematics Permutation with repetition START THE DESCRIPTION Corollary: The number of bijections from a set of elements to a set of elements is Contents Contents I Permutations: II Generalize Permutations: Definition: Definition: of a set of distinct objects is an ordered arrangment of these objects An A1.permutation ordered arrangement of r elements (or an ordered r-arrangement ) of a set is called an rAn ordered r-arrangement with repetition of the elements of the set with n elements is permutation The number of r-permutations of a set with n elements is denoted by or If , also called by an r-permutations of a set with n elements when repetition is allowed The we denoted Theorem: number of r-permutations of a set with n elements when repetition is allowed is - If is a positive integer and r is an integer with , then there are denoted by r-permutations of a set with n distinct elements Theorem: - The number of permutations of n elements: The number of r-permutations of a set of n objects with repetition allowed is Contents CONTENT 1:PREPARING KNOWLEDGE: PERMUTATIONS I Definition of permutation with repetition Contents CONTENT 2: PERMUTATION WITH REPETITION For n elements, including n1 element x1, n2 elements x2, , nk element xk The number of all repeating permutations of n elements above is: Contents permutation of the given element n Contents (n1+n2+ +nk=n) Each way of arranging n that element into n position is called a repeating Contents Contents Example 2: 3: Given With the {0;3;1;5;2;6;3;9}.4;From 5} how manymany numbers of have digits7can be so set digits A= { 1; A, how numbers digits Example 1: From X={1;2;3;4;5;6;7;8} numbers have 11exactly digits so made, which the number is present How times, eachnatural other digit is present that theofnumber set appears 1times; the number 6many appears twice; other numbers appear once andonce thatand number is divisible by 5? exactly number divisible are present once? that the number 1this is present 4istimes, otherbydigits Solution Solution Solution that the 8-digit number in which the number is present exactly times; ⇒ Suppose The number of numbers with is: digits the number with digits satisfied the first is:number is present times, the Since the number divisible by 5, 11 a8 = or Each way of 3;4 making a7 number has digits soarticle that the theCall digits 0; 2; areispresented exactly once: Because this number is divided by 5, a7 = + Case If a = 840 – 120 = 720 numbers other digits present are a repeating permutation of 11 elements The2.problem becomes the becomes calculation of thea number 6-digit numbers created from ++ Case If a8on, = 0the problem From now finding number of with digits in which the number settime; { 1;3to 3; 6;problem 9} so that the0;number appears times; number appears times is According the rules of repeating there are7the isthat present times; the digits 2; finding 3;permutation, are exactly once Atthe the becomes apresented number with digits in which the number and 3only appeared once We 3; have: presented times; digits 2; 3; 4:5 is presented exactly once According to the rules of repeat permutation there are: numbers We have: If the first digit is 0; we’ll find the number of numbers with digits in which the number There arewe 180can the2;problem is presented numbers times andthat thesatisfy number 3; appear once Therefore, make: We840 have : numbers in total 720+ = 1560 numbers Contents CONTENT : EXERCISES Contents Contents number ofaavailable Working outthe a way to human win lottery 1.Finding Protein out formation in body phone numbers No two play phonea numbers are supposed be alike If you lottery game, you maytowant to know what your odds are for winning the Proteins are formed afrom that the contain an of arrangement of amino Using phonestring-like company structures can determine number unique telephone prize permutations, acids numbers it canifissue based on the say number use For example, the lottery rules you format can winit ifwants you to pick four digits that match (e.g., The amino is important because this will0determine whether If thesequence phone should all consist of 10 digits and can use toby9, using an example of the one 1111, 9999, numbers or of 5555), youacids can work out your odds for winning a permutation protein will function not such number could be:or5653278065 calculation Defective or missing proteins cause in ways people suchslot as or sickle We canslot work out the total numbers that canserious beinavailable in this way:(since each digit0cell Each (digit position) can can be occupied 10diseases different we have to anemia position digits) can be occupied by any of the 10 digits (0 to 9) For example, insulin is atotal protein found humans.4-digit Its role is to control of = Since repetition is allowed, thisnumbers works out So, we can calculate the ofinto: possible numbers as 10 xthe 10amount x 10 x 10 sugar the numbers body so that it is neither high nor too low 10x10x10x10x10x10x10x10x10x10 =10¹⁰too numbers 10,000around different Insulin is made of 51amino acids arranged in athe specific sequence orworld permutation These a lotwe ofupcount numbers easily outnumber population of the (5 or From are there, thethat total of winning numbers (0000, 1111,2222, 3333……9999) Any that deviates from this normal sequence makes this protein billion people) whichrearrangement are 10 dysfunctional and causes diseases suchalso as diabetes However, because the phone include for area codes, So, there are 10 numbers thatnumbers can win and the totaldigits numbers you can draw are 10.000 The body has ahave mechanism to ensure that the thisright sequence is followed and =0.001 the correct they generate fewer numbers than this Therefore, you a probability of picking sequence of 10/10,000 is formed  protein In percentage terms, you have a 0.1% (0.001 x 100) chance of winning Contents CONTENT 4: SOME APPLICATIONS OF PERMUTATION THANKSFOR FOR THANKS WATCHING! WATCHING!