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, Steinerâs 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 âone 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. . (nâr+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 ânâ 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: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } In how many ways we can choose 3 men and 2 women from the room? CONTENTS iii 2.1.2 Consistency. The permutation will be = 123, 132, 213, 231, 312, 321, The number of permutations of ânâ different things taken ârâ at a time is denoted by$n_{P_{r}}$. (nâr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is − $n!â[r! Discrete Mathematics Course Notes by Drew Armstrong. .10 2.1.3 Whatcangowrong. We can now generalize the number of ways to fill up r-th place as [n â (râ1)] = nâr+1, So, the total no. Number of permutations of n distinct elements taking n elements at a time =$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 manâs 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. 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