Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. For solving these problems, mathematical theory of counting are used. There are $50/3 = 16$ numbers which are multiples of 3. / [(a_1!(a_2!) . Any subject in computer science will become much more easier after learning Discrete Mathematics . In this technique, which van Lint & Wilson (2001) call “one of the most important tools in combinatorics,” one describes a finite set X from two perspectives leading to two distinct expressions … Probability. = 6$. For instance, in how many ways can a panel of judges comprising of 6 men and 4 women be chosen from among 50 men and 38 women? Discrete Mathematics Tutorial Index Students, even possessing very little knowledge and skills in elementary arithmetic and algebra, can join our competitive mathematics classes to begin learning and studying discrete mathematics. Relation, Set, and Functions. . That means 3×4=12 different outfits. Hence, the number of subsets will be $^6C_{3} = 20$. Solution − There are 6 letters word (2 E, 1 A, 1D and 2R.) How many like both coffee and tea? . . Now, it is known as the pigeonhole principle. Graph theory. Hence, the total number of permutation is $6 \times 6 = 36$. Most basic counting formulas can be thought of as counting the number of ways to distribute either distinct or identical items to distinct recipients. . Recurrence relation and mathematical induction. Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. How many ways are there to go from X to Z? In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. . Sign up for free to create engaging, inspiring, and converting videos with Powtoon. This note explains the following topics: Induction and Recursion, Steiner’s Problem, Boolean Algebra, Set Theory, Arithmetic, Principles of Counting, Graph Theory. /Filter /FlateDecode . In a group of 50 students 24 like cold drinks and 36 like hot drinks and each student likes at least one of the two drinks. Mathematics of Master Discrete Mathematics for Computer Science with Graph Theory and Logic (Discrete Math)" today and start learning. It is a very good tool for improving reasoning and problem-solving capabilities. . There was a question on my exam which asked something along the lines of: "How many ways are there to order the letters in 'PEPPERCORN' if all the letters are used?" Next come chapters on logic, counting, and probability.We then have three chapters on graph theory: graphs, directed . Make an Impact. Discrete math. ����M>�,oX��`�N8xT����,�0�z�I�Q������������[�I9r0� '&l�v]G�q������i&��b�i� �� �`q���K�?�c�Rl Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 / 39. From his home X he has to first reach Y and then Y to Z. Discrete mathematics problem - Probability theory and counting [closed] Ask Question Asked 10 years, 6 months ago. . Problem 2 − In how many ways can the letters of the word 'READER' be arranged? For example, distributing \(k\) distinct items to \(n\) distinct recipients can be done in \(n^k\) ways, if recipients can receive any number of items, or \(P(n,k)\) ways if recipients can receive at most one item. . So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. The Basic Counting Principle. Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? (n−r+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is − $n!–[r! CONTENTS iii 2.1.2 Consistency. Starting from the 6th grade, students should some effort into studying fundamental discrete math, especially combinatorics, graph theory, discrete geometry, number theory, and discrete probability. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). in the word 'READER'. In these “Discrete Mathematics Handwritten Notes PDF”, we will study the fundamental concepts of Sets, Relations, and Functions, Mathematical Logic, Group theory, Counting Theory, Probability, Mathematical Induction, and Recurrence Relations, Graph Theory, Trees and Boolean Algebra. This tutorial includes the fundamental concepts of Sets, Relations and Functions, Mathematical Logic, Group theory, Counting Theory, Probability, Mathematical Induction, and Recurrence Relations, Graph Theory, Trees and Boolean Algebra. A permutation is an arrangement of some elements in which order matters. The Rules of Sum and Product The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. Here, the ordering does not matter. Start Discrete Mathematics Warmups. . How many different 10 lettered PAN numbers can be generated such that the first five letters are capital alphabets, the next four are digits and the last is again a capital letter. Group theory. Trees. Very Important topics: Propositional and first-order logic, Groups, Counting, Relations, introduction to graphs, connectivity, trees In how many ways we can choose 3 men and 2 women from the room? of ways to fill up from first place up to r-th-place −, $n_{ P_{ r } } = n (n-1) (n-2)..... (n-r + 1)$, $= [n(n-1)(n-2) ... (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. The different ways in which 10 lettered PAN numbers can be generated in such a way that the first five letters are capital alphabets and the next four are digits and the last is again a capital letter. The number of all combinations of n things, taken r at a time is −, $$^nC_{ { r } } = \frac { n! } If we consider two tasks A and B which are disjoint (i.e. + \frac{ n-k } { k!(n-k)! } There are n number of ways to fill up the first place. The applications of set theory today in computer science is countless. . Welcome to Discrete Mathematics 2, a course introducting Inclusion-Exclusion, Probability, Generating Functions, Recurrence Relations, and Graph Theory. )$. Counting theory. Problem 3 − In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? . Discrete Mathematics (c)Marcin Sydow Productand SumRule Inclusion-Exclusion Principle Pigeonhole Principle Permutations Generalised Permutations andCombi-nations Combinatorial Proof Binomial Coefficients DiscreteMathematics Counting (c)MarcinSydow Discrete mathematics is the branch of mathematics dealing with objects that can consider only distinct, separated values. . From there, he can either choose 4 bus routes or 5 train routes to reach Z. Pascal's identity, first derived by Blaise Pascal in 17th century, states that the number of ways to choose k elements from n elements is equal to the summation of number of ways to choose (k-1) elements from (n-1) elements and the number of ways to choose elements from n-1 elements. }$$. If each person shakes hands at least once and no man shakes the same man’s hand more than once then two men took part in the same number of handshakes. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. After filling the first place (n-1) number of elements is left. Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. It is essential to understand the number of all possible outcomes for a series of events. Set theory is a very important topic in discrete mathematics . . From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. �d�$�̔�=d9ż��V��r�e. .10 2.1.3 Whatcangowrong. This is a course note on discrete mathematics as used in Computer Science. . . Hence, there are (n-2) ways to fill up the third place. I'm taking a discrete mathematics course, and I encountered a question and I need your help. How many integers from 1 to 50 are multiples of 2 or 3 but not both? . Hence, there are 10 students who like both tea and coffee. Hence, there are (n-1) ways to fill up the second place. . Some of the discrete math topic that you should know for data science sets, power sets, subsets, counting functions, combinatorics, countability, basic proof techniques, induction, ... Information theory is also widely used in math for data science. Topics covered includes: Mathematical logic, Set theory, The real numbers, Induction and recursion, Summation notation, Asymptotic notation, Number theory, Relations, Graphs, Counting, Linear algebra, Finite fields. \dots (a_r!)]$. Discrete Mathematics Course Notes by Drew Armstrong. The cardinality of the set is 6 and we have to choose 3 elements from the set. He may go X to Y by either 3 bus routes or 2 train routes. So, Enroll in this "Mathematics:Discrete Mathematics for Computer Science . Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. Viewed 4k times 2. . Now we want to count large collections of things quickly and precisely. Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). . There are $50/6 = 8$ numbers which are multiples of both 2 and 3. Solution − From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). For choosing 3 students for 1st group, the number of ways − $^9C_{3}$, The number of ways for choosing 3 students for 2nd group after choosing 1st group − $^6C_{3}$, The number of ways for choosing 3 students for 3rd group after choosing 1st and 2nd group − $^3C_{3}$, Hence, the total number of ways $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. After filling the first and second place, (n-2) number of elements is left. If there are only a handful of objects, then you can count them with a moment's thought, but the techniques of combinatorics can extend to quickly and efficiently tabulating astronomical quantities. . Ten men are in a room and they are taking part in handshakes. (n−r+1)! >> (\frac{ k } { k!(n-k)! } From a set S ={x, y, z} by taking two at a time, all permutations are −, We have to form a permutation of three digit numbers from a set of numbers $S = \lbrace 1, 2, 3 \rbrace$. For solving these problems, mathematical theory of counting are used. + \frac{ (n-1)! } . Active 10 years, 6 months ago. . What is Discrete Mathematics Counting Theory? { r!(n-r)! Thank you. There must be at least two people in a class of 30 whose names start with the same alphabet. x��X�o7�_�G����Ozm�+0�m����\����d��GJG�lV'H�X�-J"$%J�`K&���8���8�i��ז�Jq��6�~��lғ)W,�Wl�d��gRmhVL���`.�L���N~�Efy�*�n�ܢ��ޱߧ?��z�������`|$�I��-��z�o���X�� ���w�]Lsm�K��4j�"���#gs$(�i5��m!9.����63���Gp�hЉN�/�&B��;�4@��J�?n7 CO��>�Ytw�8FqX��χU�]A�|D�C#}��kW��v��G �������m����偅^~�l6��&) ��J�1��v}�â�t�Wr���k��U�k��?�d���B�n��c!�^Հ�T�Ͳm�х�V��������6�q�o���Юn�n?����˳���x�q@ֻ[ ��XB&`��,f|����+��M`#R������ϕc*HĐ}�5S0H A combination is selection of some given elements in which order does not matter. %���� Chapter 1 Counting ¶ One of the first things you learn in mathematics is how to count. If there are n elements of which $a_1$ are alike of some kind, $a_2$ are alike of another kind; $a_3$ are alike of third kind and so on and $a_r$ are of $r^{th}$ kind, where $(a_1 + a_2 + ... a_r) = n$. . If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. Today we introduce set theory, elements, and how to build sets.This video is an updated version of the original video released over two years ago. There are 6 men and 5 women in a room. . 70 0 obj << Closed. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. (1!)(1!)(2!)] . So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. . . The number of ways to choose 3 men from 6 men is $^6C_{3}$ and the number of ways to choose 2 women from 5 women is $^5C_{2}$, Hence, the total number of ways is − $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. Proof − Let there be ‘n’ different elements. { k!(n-k-1)! Boolean Algebra. . Notes on Discrete Mathematics by James Aspnes. Question − A boy lives at X and wants to go to School at Z. In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. $A \cap B = \emptyset$), then mathematically $|A \cup B| = |A| + |B|$, The Rule of Product − If a sequence of tasks $T_1, T_2, \dots, T_m$ can be done in $w_1, w_2, \dots w_m$ ways respectively and every task arrives after the occurrence of the previous task, then there are $w_1 \times w_2 \times \dots \times w_m$ ways to perform the tasks. Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations . . The Rule of Sum − If a sequence of tasks $T_1, T_2, \dots, T_m$ can be done in $w_1, w_2, \dots w_m$ ways respectively (the condition is that no tasks can be performed simultaneously), then the number of ways to do one of these tasks is $w_1 + w_2 + \dots +w_m$. . . �.����2�(�^�� 㣯`U��$Nn$%�u��p�;�VY�����W��}����{SH�W���������-zHLJ�f� R'����;���q��Y?���?�WX���:5(�� �3a���Ã*p0�4�V����y�g�q:�k��F�̡[I�6)�3G³R�%��, %Ԯ3 Would this be 10! . . 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