# counting theory 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. It is a very good tool for improving reasoning and problem-solving capabilities. (nâr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is −$n!â[r! 70 0 obj << Trees. . $A \cap B = \emptyset$), 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. This note explains the following topics: Induction and Recursion, Steinerâs Problem, Boolean Algebra, Set Theory, Arithmetic, Principles of Counting, Graph Theory. Set theory is a very important topic in discrete mathematics . . /Length 1123 For solving these problems, mathematical theory of counting are used. 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. . . From there, he can either choose 4 bus routes or 5 train routes to reach Z. Counting theory. The Rules of Sum and Product The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. . Active 10 years, 6 months ago. . Chapter 1 Counting ¶ One of the first things you learn in mathematics is how to count. . Hence, there are 10 students who like both tea and coffee. { (k-1)!(n-k)! } Example: There are 6 flavors of ice-cream, and 3 different cones. . Graph theory. . The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. Hence, the total number of permutation is $6 \times 6 = 36$. Relation, Set, and Functions. /Filter /FlateDecode }$,$= (n-1)! + \frac{ n-k } { k!(n-k)! } Now, it is known as the pigeonhole principle. Next come chapters on logic, counting, and probability.We then have three chapters on graph theory: graphs, directed 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. After filling the first and second place, (n-2) number of elements is left. 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 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. . 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. That means 3×4=12 different outfits. . . 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). 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. Mathematics of Master Discrete Mathematics for Computer Science with Graph Theory and Logic (Discrete Math)" today and start learning. \dots (a_r!)]$. { r!(n-r)! Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. . Would this be 10! �.����2�(�^�� 㣯U��$Nn$%�u��p�;�VY�����W��}����{SH�W���������-zHLJ�f� R'����;���q��Y?���?�WX���:5(�� �3a���Ã*p0�4�V����y�g�q:�k��F�̡[I�6)�3G³R�%��, %Ԯ3 The Basic Counting Principle. / [(a_1!(a_2!) ����M>�,oX���N8xT����,�0�z�I�Q������������[�I9r0� '&l�v]G�q������i&��b�i� �� �q���K�?�c�Rl Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. Boolean Algebra. Hence, there are (n-2) ways to fill up the third place. . If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. material, may be used as a textbook for a formal course in discrete mathematics or as a supplement to all current texts. What is Discrete Mathematics Counting Theory? . = 6$. )$. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. . . Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations The applications of set theory today in computer science is countless. Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. 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. }. %���� .10 2.1.3 Whatcangowrong. + \frac{ (n-1)! } . Most basic counting formulas can be thought of as counting the number of ways to distribute either distinct or identical items to distinct recipients. (1!)(1!)(2!)] /\: [(2!) Question − A boy lives at X and wants to go to School at Z. 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. Solution − As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! There are$50/3 = 16$numbers which are multiples of 3. In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. Problem 3 − In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? How many like both coffee and tea? There are n number of ways to fill up the first place. The cardinality of the set is 6 and we have to choose 3 elements from the set. . This is a course note on discrete mathematics as used in Computer Science. The remaining 3 vacant places will be filled up by 3 vowels in$^3P_{3} = 3! In other words a Permutation is an ordered Combination of elements. Hence, there are (n-1) ways to fill up the second place. 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