Solution − There are 3 vowels and 3 consonants in the word 'ORANGE'. 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. CONTENTS iii 2.1.2 Consistency. 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. 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. . In how many ways we can choose 3 men and 2 women from the room? $A \cap B = \emptyset$), 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. After filling the first place (n-1) number of elements is left. Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. �.����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 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. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. Discrete Mathematics Course Notes by Drew Armstrong. Problem 1 − From a bunch of 6 different cards, how many ways we can permute it? It is a very good tool for improving reasoning and problem-solving capabilities. It is increasingly being applied in the practical fields of mathematics and computer science. . Thank you. / [(a_1!(a_2!) ]$, The number of circular permutations of n different elements taken x elements at time = $^np_{x}/x$, The number of circular permutations of n different things = $^np_{n}/n$. So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. From his home X he has to first reach Y and then Y to Z. Ten men are in a room and they are taking part in handshakes. . 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. Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). The permutation will be $= 6! 2 CS 441 Discrete mathematics for CS M. Hauskrecht Basic counting rules • Counting problems may be hard, and easy solutions are not obvious • Approach: – simplify the solution by decomposing the problem • Two basic decomposition rules: – Product rule • A count decomposes into a sequence of dependent counts /\: [(2!) Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. From there, he can either choose 4 bus routes or 5 train routes to reach Z. Make an Impact. Any subject in computer science will become much more easier after learning Discrete Mathematics . So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. . In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. Question − A boy lives at X and wants to go to School at Z. Different three digit numbers will be formed when we arrange the digits. Hence, the number of subsets will be $^6C_{3} = 20$. (n−r+1)! 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. Relation, Set, and Functions. Set theory is a very important topic in discrete mathematics . = 6$ ways. = 720$. + \frac{ (n-1)! } . . How many integers from 1 to 50 are multiples of 2 or 3 but not both? Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } For two sets A and B, the principle states −, $|A \cup B| = |A| + |B| - |A \cap B|$, For three sets A, B and C, the principle states −, $|A \cup B \cup C | = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C |$, $|\bigcup_{i=1}^{n}A_i|=\sum\limits_{1\leq i�,oX��`�N8xT����,�0�z�I�Q������������[�I9r0� '&l�v]G�q������i&��b�i� �� �`q���K�?�c�Rl There are n number of ways to fill up the first place. . This note explains the following topics: Induction and Recursion, Steiner’s Problem, Boolean Algebra, Set Theory, Arithmetic, Principles of Counting, Graph Theory. . Today we introduce set theory, elements, and how to build sets.This video is an updated version of the original video released over two years ago. )$. . (\frac{ k } { k!(n-k)! } The permutation will be = 123, 132, 213, 231, 312, 321, The number of permutations of ‘n’ different things taken ‘r’ at a time is denoted by $n_{P_{r}}$. . . A combination is selection of some given elements in which order does not matter. In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. . How many ways are there to go from X to Z? Why one needs to study the discrete math It is essential for college-level maths and beyond that too Discrete Mathematics Tutorial Index Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. 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?" If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. Active 10 years, 6 months ago. How many ways can you choose 3 distinct groups of 3 students from total 9 students? . . Recurrence relation and mathematical induction. + \frac{ n-k } { k!(n-k)! } . For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? Problem 2 − In how many ways can the letters of the word 'READER' be arranged? . . 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. stream . When there are m ways to do one thing, and n ways to do another, then there are m×n ways of doing both. The number of all combinations of n things, taken r at a time is −, $$^nC_{ { r } } = \frac { n! } Hence, there are 10 students who like both tea and coffee. { r!(n-r)! For solving these problems, mathematical theory of counting are used. }$, $= (n-1)! The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! Hence, the total number of permutation is $6 \times 6 = 36$. /Length 1123 In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. .10 2.1.3 Whatcangowrong. { (k-1)!(n-k)! } }$$. Trees. 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. Graph theory. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. of ways to fill up from first place up to r-th-place −, $n_{ P_{ r } } = n (n-1) (n-2)..... (n-r + 1)$, $= [n(n-1)(n-2) ... (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. Solution − There are 6 letters word (2 E, 1 A, 1D and 2R.) Discrete mathematics is the branch of mathematics dealing with objects that can consider only distinct, separated values. Start Discrete Mathematics Warmups. Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. This is a course note on discrete mathematics as used in Computer Science. Problem 3 − In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? . 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 … . Example: There are 6 flavors of ice-cream, and 3 different cones. I'm taking a discrete mathematics course, and I encountered a question and I need your help. Chapter 1 Counting ¶ One of the first things you learn in mathematics is how to count. . 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. in the word 'READER'. The first three chapters cover the standard material on sets, relations, and functions and algorithms. There must be at least two people in a class of 30 whose names start with the same alphabet. Viewed 4k times 2. Probability. Next come chapters on logic, counting, and probability.We then have three chapters on graph theory: graphs, directed . Here, the ordering does not matter. /Filter /FlateDecode The Basic Counting Principle. Now, it is known as the pigeonhole principle. We can now generalize the number of ways to fill up r-th place as [n – (r–1)] = n–r+1, So, the total no. Notes on Discrete Mathematics by James Aspnes. The Rules of Sum and Product The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. Closed. 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). . Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. . �d�$�̔�=d9ż��V��r�e. material, may be used as a textbook for a formal course in discrete mathematics or as a supplement to all current texts. . Hence, there are (n-2) ways to fill up the third place. 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. 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 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. Discrete Mathematics Handwritten Notes PDF. . %PDF-1.5 The applications of set theory today in computer science is countless. After filling the first and second place, (n-2) number of elements is left. Solution − As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! Sign up for free to create engaging, inspiring, and converting videos with Powtoon. There are $50/6 = 8$ numbers which are multiples of both 2 and 3. . So, Enroll in this "Mathematics:Discrete Mathematics for Computer Science . Mathematics of Master Discrete Mathematics for Computer Science with Graph Theory and Logic (Discrete Math)" today and start learning. { k!(n-k-1)! (n−r+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is − $n!–[r! Then, number of permutations of these n objects is = $n! . = 6$. Now we want to count large collections of things quickly and precisely. 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 − $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. 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 What is Discrete Mathematics Counting Theory? The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. 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 $(a_1 + a_2 + ... a_r) = n$. Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations That means 3×4=12 different outfits. He may go X to Y by either 3 bus routes or 2 train routes. Hence, there are (n-1) ways to fill up the second place. . . Below, you will find the videos of each topic presented. . . 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 . Number of permutations of n distinct elements taking n elements at a time = $n_{P_n} = n!$, The number of permutations of n dissimilar elements taking r elements at a time, when x particular things always occupy definite places = $n-x_{p_{r-x}}$, The number of permutations of n dissimilar elements when r specified things always come together is − $r! There are $50/3 = 16$ numbers which are multiples of 3. 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