# counting theory discrete math

• December 24th, 2020
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How many ways are there to go from X to Z? %���� . material, may be used as a textbook for a formal course in discrete mathematics or as a supplement to all current texts. From his home X he has to first reach Y and then Y to Z. 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. Would this be 10! Make an Impact. + \frac{ (n-1)! } Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. . 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 $n_{P_{r}}$. 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. Here, the ordering does not matter. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. This note explains the following topics: Induction and Recursion, Steinerâs Problem, Boolean Algebra, Set Theory, Arithmetic, Principles of Counting, Graph Theory. ����M>�,oX���N8xT����,�0�z�I�Q������������[�I9r0� '&l�v]G�q������i&��b�i� �� �q���K�?�c�Rl )$. .10 2.1.3 Whatcangowrong. Solution − As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be$^6P_{6} = 6! What is Discrete Mathematics Counting Theory? Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. 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. . Hence, there are (n-2) ways to fill up the third place. Boolean Algebra. 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 − $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. 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¦ That means 3×4=12 different outfits. For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? Hence, the total number of permutation is $6 \times 6 = 36$. 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. Recurrence relation and mathematical induction. �d�$�̔�=d9ż��V��r�e. Proof − Let there be ânâ different elements. There are 6 men and 5 women in a room. }$$. �.����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 . . Most basic counting formulas can be thought of as counting the number of ways to distribute either distinct or identical items to distinct recipients. Very Important topics: Propositional and first-order logic, Groups, Counting, Relations, introduction to graphs, connectivity, trees . There must be at least two people in a class of 30 whose names start with the same alphabet.$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 permutation will be$= 6! CONTENTS iii 2.1.2 Consistency. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. 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. /\: [(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]$. 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$. 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$. It is increasingly being applied in the practical fields of mathematics and computer science. . Discrete Mathematics Course Notes by Drew Armstrong. . A combination is selection of some given elements in which order does not matter. Then, number of permutations of these n objects is = $n! Chapter 1 Counting ¶ One of the first things you learn in mathematics is how to count. 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 . stream Solution − There are 6 letters word (2 E, 1 A, 1D and 2R.) 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. How many like both coffee and tea? Example: you have 3 shirts and 4 pants. Probability. In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. There are$50/6 = 8$numbers which are multiples of both 2 and 3. . The Basic Counting Principle. Notes on Discrete Mathematics by James Aspnes. Thank you. >> Discrete mathematics problem - Probability theory and counting [closed] Ask Question Asked 10 years, 6 months ago. Different three digit numbers will be formed when we arrange the digits. 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�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 He may go X to Y by either 3 bus routes or 2 train routes. (\frac{ k } { k!(n-k)! } 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. . Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. (1!)(1!)(2!)] . A permutation is an arrangement of some elements in which order matters. 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$. (nâr+1)! Problem 1 − From a bunch of 6 different cards, how many ways we can permute it? Solution − From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). Solution − There are 3 vowels and 3 consonants in the word 'ORANGE'. Ten men are in a room and they are taking part in handshakes. Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). = 720$. The applications of set theory today in computer science is countless. 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. Why one needs to study the discrete math It is essential for college-level maths and beyond that too How many integers from 1 to 50 are multiples of 2 or 3 but not both? 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! Discrete mathematics is the branch of mathematics dealing with objects that can consider only distinct, separated values. It is a very good tool for improving reasoning and problem-solving capabilities. Welcome to Discrete Mathematics 2, a course introducting Inclusion-Exclusion, Probability, Generating Functions, Recurrence Relations, and Graph Theory. . Counting theory. Set theory is a very important topic in discrete mathematics . 70 0 obj << 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 Handwritten Notes PDF. /Length 1123 . 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. . }$,$= (n-1)! Sign up for free to create engaging, inspiring, and converting videos with Powtoon. . ]$, The number of circular permutations of n different elements taken x elements at time =$^np_{x}/x$, The number of circular permutations of n different things =$^np_{n}/n$. . This is a course note on discrete mathematics as used in Computer Science. 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. There are$50/3 = 16$numbers which are multiples of 3. Mathematically, for any positive integers k and n:$^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$,$= \frac{ (n-1)! } Problem 2 − In how many ways can the letters of the word 'READER' be arranged? . Group theory. Active 10 years, 6 months ago. Hence, there are 10 students who like both tea and coffee. There are n number of ways to fill up the first place. How many ways can you choose 3 distinct groups of 3 students from total 9 students? . Closed. In other words a Permutation is an ordered Combination of elements. After filling the first and second place, (n-2) number of elements is left. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. . The Rules of Sum and Product The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. = 6$. 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. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. Mathematics of Master Discrete Mathematics for Computer Science with Graph Theory and Logic (Discrete Math)" today and start learning. = 180.$. Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. Relation, Set, and Functions. Discrete math. For solving these problems, mathematical theory of counting are used. . So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. in the word 'READER'. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. . . { (k-1)!(n-k)! } . . Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). \dots (a_r!)]$. Pigeonhole Principle states that if there are fewer pigeon holes than total number of pigeons and each pigeon is put in a pigeon hole, then there must be at least one pigeon hole with more than one pigeon. Some of the discrete math topic that you should know for data science sets, power sets, subsets, counting functions, combinatorics, countability, basic proof techniques, induction, ... Information theory is also widely used in math for data science. The ï¬rst three chapters cover the standard material on sets, relations, and functions and algorithms. The remaining 3 vacant places will be filled up by 3 vowels in$^3P_{3} = 3! The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. /Filter /FlateDecode Trees. Now we want to count large collections of things quickly and precisely. . (nâr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is −$n!â[r! %PDF-1.5 . Below, you will find the videos of each topic presented. . For solving these problems, mathematical theory of counting are used. Now, it is known as the pigeonhole principle. In how many ways we can choose 3 men and 2 women from the room? . . So, Enroll in this "Mathematics:Discrete Mathematics for Computer Science . It is essential to understand the number of all possible outcomes for a series of events. Hence, the number of subsets will be $^6C_{3} = 20$. 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. . The cardinality of the set is 6 and we have to choose 3 elements from the set. 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