Here \ discrete as opposed to continuous typically also means nite, although we will consider some in nite structures as well. Ralston, discrete algorithmic mathematics, a k peters, 3 rd edition, 2004. Counting poker hands provides multiple additional examples. Practice counting possible outcomes in a variety of situations. We wrap up the section on counting by doing a few practice problems and showing the intuitions behind solving each problem. Balakrishnan, theory and probl ems of combinatorics, schaums outline series, mcgrawhill, 1995 s. Discrete mathematics deals with objects that come in discrete bundles, e. Rule of sum rule of product counting integers in a range the rule of sum and the rule of product are two basic principles of counting that are. Discrete mathematics counting theory in discrete mathematics.
Combination problems and solutions counting principle. Discrete mathematics and its applications lecture 3. For example, with luck, you just stated the sum principle illustrated in figure 1. We introduce some elementary concepts of combinatorics in terms of counting problems. Counting principle algebra 2, discrete mathematics and. Combinatorics is the study of arrangements of objects, it is an important part of discrete mathematics. Besides reading the book, students are strongly encouraged to do all the. When there are m ways to do one thing, and n ways to do another, then there are m. Here are two more, straightforward examples where the question is to count the steps of. Discrete mathematics rule of sum and rule of product.
How many bitstrings of length 8 start either with a bit 1 or end with. In contrast, continuous mathematics deals with objects that vary continuously, e. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. Discrete here is used as the opposite of continuous. Chapter 4 counting principles discrete mathematics and. Lets come up with an algorithm that generates a seating. Browse other questions tagged combinatorics discretemathematics or. Cs 70 discrete mathematics and probability theory counting.
Discrete mathematics counting theory tutorialspoint. Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. Discrete mathematics and probability algebra 2 mathplanet. The basics of counting the pigeonhole principle permutations and combinations binomial coef. The aim of this book is not to cover discrete mathematics in. Hopefully this short introduction will shed some light on what the subject is about and what you can expect as you move. This page is dedicated to problem solving on the notions of rule of sum also known as addition principle and rule of product also known as multiplication principle. Dec 21, 2018 we introduce the rule of sum addition rule and rule of product product rule in counting. Home highlights for high school mathematics combinatorics. Number of available phone numbers with 7 digits in the local calling area number of possible match starters football, basketball given. Suppose that i love discrete math tshirts come in five different.
This is a consequence of the pigeonhole principle, but the fact. Rosen, discrete mathematics and its applications somchai prasitjutragul, discrete mathematics. Dec 21, 2014 we wrap up the section on counting by doing a few practice problems and showing the intuitions behind solving each problem. The fine art of counting lecture notes lecture notes course home. Counting theory discrete mathematics in daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. Discrete mathematics counting theory in daily lives, many a times one needs to. Problems concerning a deck of cards are also good examples. There are nd ways to do a task if it can be done using a procedure that can be carried out in n ways, and for every way w, exactly d of the n ways correspond to way w. For solving these problems, mathematical theory of counting are used. We now present three simple examples to illustrate this. Menu algebra 2 discrete mathematics and probability counting principle. The existence, enumeration, analysis and optimization of discrete structures. Principle of inclusionexclusion in general, we have the following rule.
More formally, we begin with a basic counting principle. Discrete mathematics counting theory in daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. Menu algebra 2 discrete mathematics and probability counting principle if we are dealing with the occurrence of more than one event or activity, sometimes it is important to be able to determine how many possible outcomes exist. Find materials for this course in the pages linked along the left. Hauskrecht counting assume we have a set of objects with certain properties counting is used to determine the number of these objects examples. We follow a highlevel approach also adopted in most introductory textbooks in discrete mathematics as long it is well understood how we can technically formalize the arguments. Mathematical logic, set theory, the real numbers, induction and recursion, summation notation, asymptotic notation, number theory, relations, graphs, counting, linear algebra, finite fields. Discrete structures lecture notes stanford university. Think of digital watches versus analog watches ones where the second hand loops around continuously without stopping. One way to formally introduce counting principles is through countable sets. Ma8351 notes discrete mathematics to extend students logical and mathematical maturity and ability to deal with abstraction. Lecture notes principles of discrete applied mathematics. These problems contain among them the kernels of many of the fundamental ideas of combinatorics.
This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Pigeonhole principle states that if there are fewer pigeon holes than total. The basics of counting the pigeonhole principle permutations and combinations binomial coefcients and identities generalized permutations and combinations colin stirling informatics discrete mathematics chapter 6 today 2 39. Chapter 4 counting principles learning objectives on completing this chapter, you should be able to. The mathematics in these applications is collectively called discrete mathematics. We must count objects to solve many different types of problems, like the determining whether there are enough telephone numbers or internet protocal ip addresses to meet demand. The aim of this book is not to cover discrete mathematics in depth it should be clear. Interconnections, generalizations and specializationrelations between sev. 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. If we are dealing with the occurrence of more than one event or activity, sometimes it is important to be able to determine how many possible outcomes exist.
If we can break the objects we are counting into separate, nonoverlapping disjoint cases, the total number of objects is the sum of the numbers for each individual case. Here \discrete as opposed to continuous typically also means nite, although we will consider some in nite structures as well. Discrete mathematics rule of sum and rule of product duration. The discrete mathematics notes pdf dm notes pdf book starts with the topics covering logic and proof, strong induction,pigeon hole principle, isolated vertex, directed graph, alebric structers, lattices and boolean algebra, etc. The rule of sum and rule of product are used to decompose difficult counting problems into simple problems.
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. From a set s x, y, z by taking two at a time, all permutations are. To solve problems on this page, you should be familiar with the following notions. For n given weights, what is the greatest number of different. Discrete mathematics pdf notes dm lecture notes pdf. Principle of inclusionexclusion how many bit strings of length eight either start with 1 or end with 00. Below, we give two examples of proofs by contradiction. Discrete mathematics counting saad mneimneh 1 n choose k consider the problem of seating n people on n chairs. The rule of sum addition principle and the rule of product multiplication principle are stated as below. Ma8351 notes discrete mathematics regulation 2017 anna. Basic counting department of computer science university of. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. Hauskrecht inclusionexclusion principle used in counts where the decomposition yields two count tasks with overlapping elements if we used the sum rule some elements would be counted twice inclusionexclusion principle. Throughout the course, students will be expected to demonstrate their understanding of discrete mathematics by being able to do each of the following.
Cs 70 discrete mathematics and probability theory fall 2009 satish rao,david tse note 9 counting in the next major topic of the course, we will be looking at probability. For a pair of sets a and b, a b denotes theircartesian product. Rule of sum and rule of product problem solving brilliant. Working from basic principles and using elementary tools we develop the basic theory in its full generality. In stage 1 we choose a person for chair 1, in stage 2 we choose a person for chair 2, etc. We introduce the rule of sum addition rule and rule of product product rule in counting. It is essential to understand the number of all possible outcomes for a series of events. Discrete mathematics notes ma8351 pdf free download. Ma8351 notes discrete mathematics regulation 2017 anna university free download.
Counting onetoone functions i how manyonetoonefunctions are there from a set with 3 elements to a set with 5 elements. Share copy and redistribute the material in any medium or format adapt remix, transform, and build upon the material under the following terms. Discrete mathematics cmarcin sydow productand sumrule inclusionexclusion principle pigeonhole principle permutations generalised permutations andcombinations combinatorial proof binomial coe. This course acts as an introduction to contemporary discrete mathematics. Lecture notes for college discrete mathematics download book. Discrete mathematics counting practice we wrap up the section on counting by doing a few practice problems and showing the intuitions behind solving each problem. These problems cover everything from counting the number of ways to get dressed in the morning to counting the number of ways to build a custom pizza. Think of digital watches versus analog watches ones where the. This is a course note on discrete mathematics as used in computer science. Roughly, the plan is to touch on the following topics. Along the way, we recall some notions from discrete mathematics that you are.
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