counting theory discrete math

Different three digit numbers will be formed when we arrange the digits. Trees. Chapter 1 Counting ¶ One of the first things you learn in mathematics is how to count. \dots (a_r!)]$. In other words a Permutation is an ordered Combination of elements. . (nâr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is − $n!â[r! If each person shakes hands at least once and no man shakes the same manâs hand more than once then two men took part in the same number of handshakes. . Hence, there are (n-2) ways to fill up the third place. ]$, 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 = $^np_{n}/n$. Discrete Mathematics Course Notes by Drew Armstrong. How many ways can you choose 3 distinct groups of 3 students from total 9 students? Discrete Mathematics (c)Marcin Sydow Productand SumRule Inclusion-Exclusion Principle Pigeonhole Principle Permutations Generalised Permutations andCombi-nations Combinatorial Proof Binomial Coeï¬cients DiscreteMathematics Counting (c)MarcinSydow It is increasingly being applied in the practical fields of mathematics and computer science. material, may be used as a textbook for a formal course in discrete mathematics or as a supplement to all current texts. In this technique, which van Lint & Wilson (2001) call âone of the most important tools in combinatorics,â one describes a finite set X from two perspectives leading to two distinct expressions â¦ = 6$ ways. { r!(n-r)! . /\: [(2!) We can now generalize the number of ways to fill up r-th place as [n â (râ1)] = nâr+1, So, the total no. The applications of set theory today in computer science is countless. 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. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. In these âDiscrete Mathematics Handwritten Notes PDFâ, 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. %PDF-1.5 = 6$. Counting theory. . 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? %�� This is a course note on discrete mathematics as used in Computer Science. Relation, Set, and Functions. How many integers from 1 to 50 are multiples of 2 or 3 but not both? of ways to fill up from first place up to r-th-place −, $n_{ P_{ r } } = n (n-1) (n-2)..... (n-r + 1)$, $= [n(n-1)(n-2) ... (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. After filling the first and second place, (n-2) number of elements is left. 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. Then, number of permutations of these n objects is = $n! There must be at least two people in a class of 30 whose names start with the same alphabet. Solution − From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). Graph theory. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. (nâr+1)! The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! Solution − As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! When there are m ways to do one thing, and n ways to do another, then there are m×n ways of doing both. The different ways in which 10 lettered PAN numbers can be generated in such a way that the first five letters are capital alphabets and the next four are digits and the last is again a capital letter. Ten men are in a room and they are taking part in handshakes. . )$. in the word 'READER'. 2 CS 441 Discrete mathematics for CS M. Hauskrecht Basic counting rules â¢ Counting problems may be hard, and easy solutions are not obvious â¢ Approach: â simplify the solution by decomposing the problem â¢ Two basic decomposition rules: â Product rule â¢ A count decomposes into a sequence of dependent counts That means 3×4=12 different outfits. Discrete Mathematics Tutorial Index . 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?" . The Basic Counting Principle. = 180.$. . The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. . Start Discrete Mathematics Warmups. . x��X�o7�_�G��Ozm�+0�m��\��d��GJG�lV'H�X�-J"$%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 Below, you will find the videos of each topic presented. (1!)(1!)(2!)] Any subject in computer science will become much more easier after learning Discrete Mathematics . . 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 $w_1 + w_2 + \dots +w_m$. Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. 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. 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. 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 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. For solving these problems, mathematical theory of counting are used. Boolean Algebra. Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? 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. . Problem 1 − From a bunch of 6 different cards, how many ways we can permute it? 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. It is a very good tool for improving reasoning and problem-solving capabilities. . Most basic counting formulas can be thought of as counting the number of ways to distribute either distinct or identical items to distinct recipients. /Filter /FlateDecode . . Now we want to count large collections of things quickly and precisely. A combination is selection of some given elements in which order does not matter. If we consider two tasks A and B which are disjoint (i.e. stream { (k-1)!(n-k)! } Why one needs to study the discrete math It is essential for college-level maths and beyond that too 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. 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. Sign up for free to create engaging, inspiring, and converting videos with Powtoon. . + \frac{ (n-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. �.��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 . What is Discrete Mathematics Counting Theory? Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } Thank you. Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). (\frac{ k } { k!(n-k)! } Discrete mathematics problem - Probability theory and counting [closed] Ask Question Asked 10 years, 6 months ago. Make an Impact. 70 0 obj << . So, Enroll in this "Mathematics:Discrete Mathematics for Computer Science . From his home X he has to first reach Y and then Y to Z. . From there, he can either choose 4 bus routes or 5 train routes to reach Z. There are $50/6 = 8$ numbers which are multiples of both 2 and 3. Now, it is known as the pigeonhole principle. Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. I'm taking a discrete mathematics course, and I encountered a question and I need your help. 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. Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! The cardinality of the set is 6 and we have to choose 3 elements from the set. = 720$. How many like both coffee and tea? Notes on Discrete Mathematics by James Aspnes. / [(a_1!(a_2!) . Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. Discrete Mathematics Handwritten Notes PDF. Problem 3 − In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? 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 − $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. . $A \cap B = \emptyset$), 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. The number of all combinations of n things, taken r at a time is −, $$^nC_{ { r } } = \frac { n! } Probability. In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. . There are $50/3 = 16$ numbers which are multiples of 3. }$, $= (n-1)! Starting from the 6th grade, students should some effort into studying fundamental discrete math, especially combinatorics, graph theory, discrete geometry, number theory, and discrete probability. Mathematics of Master Discrete Mathematics for Computer Science with Graph Theory and Logic (Discrete Math)" today and start learning. Discrete math. Number of permutations of n distinct elements taking n elements at a time = $n_{P_n} = n!$, The number of permutations of n dissimilar elements taking r elements at a time, when x particular things always occupy definite places = $n-x_{p_{r-x}}$, The number of permutations of n dissimilar elements when r specified things always come together is − $r! . Hence, there are (n-1) ways to fill up the second place. Solution − There are 3 vowels and 3 consonants in the word 'ORANGE'. Example: you have 3 shirts and 4 pants. }$$. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. + \frac{ n-k } { k!(n-k)! } . Would this be 10! There are n number of ways to fill up the first place. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. Example: There are 6 flavors of ice-cream, and 3 different cones. . Mastering Discrete Math ( Discrete mathematics ) is such a crucial event for any computer science engineer. CONTENTS iii 2.1.2 Consistency. Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. Next come chapters on logic, counting, and probability.We then have three chapters on graph theory: graphs, directed In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. Group theory. It is essential to understand the number of all possible outcomes for a series of events. Hence, there are 10 students who like both tea and coffee. ��M>�,oX��`�N8xT��,�0�z�I�Q��[�I9r0� '&l�v]G�q��i&��b�i� �� `q��K�?�c�Rl . This note explains the following topics: Induction and Recursion, Steinerâs Problem, Boolean Algebra, Set Theory, Arithmetic, Principles of Counting, Graph Theory. . . Hence, the number of subsets will be $^6C_{3} = 20$. Hence, the total number of permutation is $6 \times 6 = 36$. 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 $(a_1 + a_2 + ... a_r) = n$. Very Important topics: Propositional and first-order logic, Groups, Counting, Relations, introduction to graphs, connectivity, trees If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. Problem 2 − In how many ways can the letters of the word 'READER' be arranged? . . . Here, the ordering does not matter. . .10 2.1.3 Whatcangowrong. Active 10 years, 6 months ago. In how many ways we can choose 3 men and 2 women from the room? Closed. The ï¬rst three chapters cover the standard material on sets, relations, and functions and algorithms. 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. �d�$�̔�=d9ż��V��r�e. Pigeonhole Principle states that if there are fewer pigeon holes than total number of pigeons and each pigeon is put in a pigeon hole, then there must be at least one pigeon hole with more than one pigeon. How many ways are there to go from X to Z? He may go X to Y by either 3 bus routes or 2 train routes. . /Length 1123 The permutation will be = 123, 132, 213, 231, 312, 321, The number of permutations of ânâ different things taken ârâ at a time is denoted by $n_{P_{r}}$. . There are 6 men and 5 women in a room. A permutation is an arrangement of some elements in which order matters. Recurrence relation and mathematical induction. Proof − Let there be ânâ different elements. >> Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). . { k!(n-k-1)! . Viewed 4k times 2. 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 is the branch of mathematics dealing with objects that can consider only distinct, separated values. Question − A boy lives at X and wants to go to School at Z. Solution − There are 6 letters word (2 E, 1 A, 1D and 2R.) So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. Welcome to Discrete Mathematics 2, a course introducting Inclusion-Exclusion, Probability, Generating Functions, Recurrence Relations, and Graph Theory. The permutation will be $= 6! Set theory is a very important topic in discrete mathematics . 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N pigeons are put into m pigeonholes where n > m, there are $ 50/2 = +! Are n number of permutations of these n objects is = $!. = 3 X he has to first reach Y and then Y to.! Needs to study the discrete Math ( discrete Math ( discrete Math discrete... `` mathematics: discrete mathematics people who like cold drinks and Y be the set of people like. To go from X to Y, he can go in $ ^3P_ { 3 } =!... ) ( 2! ) ( 2 E, 1 a, then $ |A \times =... Many ways can the letters of the union of multiple non-disjoint sets understand the of. Two people in a class of 30 whose names start with the alphabet...