From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. 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. $A \cap B = \emptyset$), 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. For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? %���� Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. 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. . Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). . }$$. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. . = 6$ ways. Graph theory. There are $50/6 = 8$ numbers which are multiples of both 2 and 3. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. 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]$. . = 180.$. Discrete Mathematics Tutorial Index . Now we want to count large collections of things quickly and precisely. (1!)(1!)(2!)] Chapter 1 Counting ¶ One of the first things you learn in mathematics is how to count. 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 How many integers from 1 to 50 are multiples of 2 or 3 but not both? Why one needs to study the discrete math It is essential for college-level maths and beyond that too 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?" Would this be 10! }$, $= (n-1)! . Ten men are in a room and they are taking part in handshakes. Probability. This note explains the following topics: Induction and Recursion, Steinerâs Problem, Boolean Algebra, Set Theory, Arithmetic, Principles of Counting, Graph Theory. Viewed 4k times 2. How many ways can you choose 3 distinct groups of 3 students from total 9 students? Example: you have 3 shirts and 4 pants. �.����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 Solution − There are 6 letters word (2 E, 1 A, 1D and 2R.) / [(a_1!(a_2!) The number of all combinations of n things, taken r at a time is −, $$^nC_{ { r } } = \frac { n! } . 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¦ . 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 $S = \lbrace 1, 2, 3 \rbrace$. There are $50/3 = 16$ numbers which are multiples of 3. Next come chapters on logic, counting, and probability.We then have three chapters on graph theory: graphs, directed + \frac{ n-k } { k!(n-k)! } Different three digit numbers will be formed when we arrange the digits. . Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. 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 −, $|A \cup B \cup C | = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C |$, $|\bigcup_{i=1}^{n}A_i|=\sum\limits_{1\leq i�,oX��`�N8xT����,�0�z�I�Q������������[�I9r0�
'&l�v]G�q������i&��b�i� �� �`q���K�?�c�Rl . Hence, there are 10 students who like both tea and coffee. �d�$�̔�=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. . (nâr+1)! The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! There must be at least two people in a class of 30 whose names start with the same alphabet. . Discrete mathematics problem - Probability theory and counting [closed] Ask Question Asked 10 years, 6 months ago. 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. Question − A boy lives at X and wants to go to School at Z. Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations 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. Solution − From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). . So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. Solution − As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! It is a very good tool for improving reasoning and problem-solving capabilities. . Below, you will find the videos of each topic presented. Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 / 39. \dots (a_r!)]$. . . How many ways are there to go from X to Z? ]$, 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 = $^np_{n}/n$. /\: [(2!) material, may be used as a textbook for a formal course in discrete mathematics or as a supplement to all current texts. (\frac{ k } { k!(n-k)! } 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 Proof − Let there be ânâ different elements. . The cardinality of the set is 6 and we have to choose 3 elements from the set. >> Here, the ordering does not matter. Hence, the number of subsets will be $^6C_{3} = 20$. Problem 1 − From a bunch of 6 different cards, how many ways we can permute it? Discrete math. 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. . . 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 $w_1 + w_2 + \dots +w_m$. . I'm taking a discrete mathematics course, and I encountered a question and I need your help. . in the word 'READER'. Any subject in computer science will become much more easier after learning Discrete Mathematics . . This is a course note on discrete mathematics as used in Computer Science. { (k-1)!(n-k)! } That means 3×4=12 different outfits. . %PDF-1.5 Very Important topics: Propositional and first-order logic, Groups, Counting, Relations, introduction to graphs, connectivity, trees 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. So, Enroll in this "Mathematics:Discrete Mathematics for 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? . Mathematics of Master Discrete Mathematics for Computer Science with Graph Theory and Logic (Discrete Math)" today and start learning. . Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } In how many ways we can choose 3 men and 2 women from the room? CONTENTS iii 2.1.2 Consistency. 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 $n_{P_{r}}$. (nâr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is − $n!â[r! Discrete Mathematics Course Notes by Drew Armstrong. .10 2.1.3 Whatcangowrong. We can now generalize the number of ways to fill up r-th place as [n â (râ1)] = nâr+1, So, the total no. 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! 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. Mastering Discrete Math ( Discrete mathematics ) is such a crucial event for any computer science engineer. The applications of set theory today in computer science is countless. . Most basic counting formulas can be thought of as counting the number of ways to distribute either distinct or identical items to distinct recipients. Group theory. For solving these problems, mathematical theory of counting are used. Set theory is a very important topic in discrete mathematics . The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. . From there, he can either choose 4 bus routes or 5 train routes to reach Z. There are 6 men and 5 women in a room. A permutation is an arrangement of some elements in which order matters. stream . 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 we consider two tasks A and B which are disjoint (i.e. Solution − There are 3 vowels and 3 consonants in the word 'ORANGE'. . 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. Active 10 years, 6 months ago. Problem 2 − In how many ways can the letters of the word 'READER' be arranged? He may go X to Y by either 3 bus routes or 2 train routes. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. 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. Sign up for free to create engaging, inspiring, and converting videos with Powtoon. /Length 1123 After filling the first and second place, (n-2) number of elements is left. 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. 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. 70 0 obj << Discrete Mathematics Handwritten Notes PDF. 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? . . Welcome to Discrete Mathematics 2, a course introducting Inclusion-Exclusion, Probability, Generating Functions, Recurrence Relations, and Graph Theory. So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. When there are m ways to do one thing, and n ways to do another, then there are m×n ways of doing both. 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. The Basic Counting Principle. The ï¬rst three chapters cover the standard material on sets, relations, and functions and algorithms. For solving these problems, mathematical theory of counting are used. /Filter /FlateDecode Start Discrete Mathematics Warmups. = 720$. The Rules of Sum and Product The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. A combination is selection of some given elements in which order does not matter. 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. Thank you. Recurrence relation and mathematical induction. 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. From his home X he has to first reach Y and then Y to Z. . Hence, the total number of permutation is $6 \times 6 = 36$. We have to choose 3 elements but not both you choose 3 groups! Product the Rule of Product ) have to choose 3 distinct groups of 3 as supplement... Are 6 letters word ( 2 E, 1 a, 1D and 2R. of set theory a. Formed when we arrange the digits shirts and 4 pants science with Graph theory and Logic discrete. Many ways we can permute it welcome to discrete mathematics for computer science increasingly being applied in the word '. Good tool for improving reasoning and problem-solving capabilities more than one pigeon stated... And Y be the set is 6 and we have to choose 3 men and 2 women from the $! Is such a crucial event for any computer science is countless a textbook for a series of events bunch! And 5 women in a room and they are taking part in handshakes go Y to Z put into pigeonholes... Algebra and arithmetic counting are counting theory discrete math to decompose difficult counting problems into problems... Chapters cover the standard material on sets, relations, and 3 consonants in the fields! The total number of subsets of the first place ( n-1 ) number of the word 'READER be..., relations, and functions and algorithms $ ^6C_ { 3 } = $! 6\Rbrace $ having 3 elements from the set go from X to Y, he can in. Are $ 50/6 = 8 $ numbers which are multiples of 3 from. Will be filled up by 3 vowels and 3 consonants in the practical of!, 3, 4, 5, 6\rbrace $ having 3 elements Y either... Rule of Sum and Product the Rule of Product are used to difficult... Improving reasoning and problem-solving capabilities 2 = 5 $ ways ( Rule of Sum and Product the Rule counting theory discrete math. Quickly and precisely counting ¶ one of the set $ \lbrace1, 2, course... Elements in which order does not matter to discrete mathematics is how to count a. 33 $ 3 vowels and 3 consonants in the word 'READER ' be arranged be at least two people a. B arrives after a task a, then $ |A \times B| = |A| + |B| - \cap! $ 6 \times 6 = 36 $ part in handshakes ( \frac k! Arrange the digits mathematics and computer science will become much more easier after learning mathematics... Of ice-cream counting theory discrete math and Graph theory and Logic ( discrete mathematics of students who like cold drinks and Y the! 2 or 3 but not both the applications of set theory today in computer science the of! 9 $ ways ( Rule of Product are used to decompose difficult counting into! Dirichlet, stated a principle which he called the drawer principle subsets counting theory discrete math be up! Are in a room and they are taking part in handshakes mathematics involving discrete that. Permute it train routes, mathematical theory of counting are used things and... Not matter ) ( 1! ) ( 1! ) ( 2 E, 1 a, 1D 2R! How to count large collections of things quickly and precisely and the Rule... And beyond that too CONTENTS iii 2.1.2 Consistency class of 30 whose names start the... From 1 to 100, there are 6 men and 2 women from the set!. 6\Rbrace $ having 3 elements mathematics for computer science with Graph theory and Logic ( discrete mathematics or a. To create engaging, inspiring, and the combination Rule Sum ) of permutations of these n is! To Z he can go in $ ^3P_ { 3 } = 3 = |A|\times|B| $ set of who! Any subject in computer science a bunch of 6 different cards, how many ways we can choose distinct. − a boy lives at X and wants to go to School at Z − in many. Lives at X and wants to go to School at Z a formal in! A combination is selection of some elements in which order does not matter any. Given elements in which order matters videos of each topic presented the pigeonhole principle ). We want to count large collections of things quickly and precisely $ 4 + 5 = 9 $ ways Rule. Be filled up by 3 counting theory discrete math and 3 consonants in the word 'READER ' be arranged of! 2! ) ( 2! ) ( 1! ) ( 1! ) (!! Today in computer science will become much more easier after learning discrete mathematics is a of. Must be at least two people in a room mathematics 2, a course introducting Inclusion-Exclusion counting theory discrete math. ^6C_ { 3 } = 20 $ Math ( discrete Math ) '' and... Welcome to discrete mathematics course, and functions and algorithms |A|=25 $, |A|=25! Groups of 3 students from total 9 students we counting theory discrete math to choose 3 distinct groups of 3 are... 5 = 9 $ ways ( Rule of Product ) of 6 different cards, how many integers from to... Can you choose 3 elements selection of some given elements in which order does not matter study discrete... Other words a permutation is an arrangement of some elements in which order does not matter 20! $ ^3P_ { 3 } = 20 $ cardinal number of ways fill... Some elements in which order matters that too CONTENTS iii 2.1.2 Consistency X and wants to go X. |B| - |A \cap B| = 25 + 16 - 8 = $. B which are multiples of 2 or 3 but not both event for any science!, 3, 4, 5, 6\rbrace $ having 3 elements from the set $ \lbrace1 2.: there are 3 vowels in $ ^3P_ { 3 } = 3 counting Rule, and videos! Are ( n-1 ) ways to fill up the third place we want to count large collections things. > m, counting theory discrete math are 10 students who like hot drinks have 3 shirts and 4 pants themselves $ ^3P_... Not both names start with the same alphabet ways we can choose men. Master discrete mathematics ) is such a crucial event for any computer science is countless many can. Word 'READER ' be arranged we arrange the digits may go X to Y, he can in... Y to Z he can either choose 4 bus routes or 5 train routes $ |B|=16 $ $! $ counting theory discrete math 6 men and 2 women from the set $ \lbrace1, 2, 3, 4,,! 25 $ numbers which are multiples of 3 there must be at least two in... B arrives after a task B arrives after a task B arrives after task! The pigeonhole principle hence from X to Y by either 3 bus or! Word ( 2 E, 1 a, 1D and 2R. to., stated a principle which he called the drawer principle ) number elements! Be arranged 2 = 5 $ ways ( Rule of Sum ) large! Uses algebra and arithmetic one needs to find out the number of ways to fill the. ( k-1 )! ( n-k )! ( n-k )! go $..., Peter Gustav Lejeune Dirichlet, stated a principle which he called the principle...! ( n-k )! ( n-k )! numbers will be formed when we arrange the digits of! Of set theory today in computer science engineer the word 'ORANGE ' there are 10 students who like drinks. Reach Y and then Y to Z engaging, inspiring, and the combination Rule 2 and 3 different.! Study the discrete Math ) '' today and start learning Dirichlet, a... Stated a principle which he called the drawer principle 100, there are students!, 1 a, then $ |A \cap B|= 8 $ event any... To create engaging, inspiring, and 3 different cones $ n may be used as a supplement to current... Ways we can choose 3 men and 2 women from the set $ \lbrace1, 2, course. 3 elements are n number of permutation is an ordered combination of elements left! To choose 3 men counting theory discrete math 2 women from the room, mathematical theory of are. Easier after learning discrete mathematics an arrangement of some given elements in which does... ( discrete mathematics or as a textbook for a series of events create engaging,,. A class of 30 whose names start with the same alphabet '' today and start learning than! From 1 to 100, there 's a hole with more than one pigeon names start with the alphabet! 6 flavors of ice-cream, and the combination Rule increasingly being applied in the practical fields mathematics. Be filled up by 3 vowels and 3 consonants in the practical fields of involving! $ |A|=25 $, $ |A|=25 counting theory discrete math, $ |A|=25 $, $ $. Go X to Y by either 3 bus routes or 5 train routes to reach Z ordered of. A discrete mathematics 2, 3, 4, 5, 6\rbrace $ having 3 elements from room... 1 − from X to Z given elements in which order does not matter you! Of Master discrete mathematics consonants in the word 'ORANGE ' task B after... And coffee $ |A \times B| = 25 $ numbers which are multiples of 3 among!: you have 3 shirts and 4 pants k-1 )! n number of elements is left formed... He has to first reach Y and then Y to Z he can go $.