16-Counting Techniques

# 16-Counting Techniques - EECS 210 Discrete Structures David...

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Counting Techniques David O. Johnson EECS 210 (Fall 2016) 1 EECS 210 Discrete Structures David O. Johnson Fall 2016

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Reminders Homework 4 due: Thursday, October 27 (today) Connect 4 due: 11:59 PM, Thursday, November 3 Counting Techniques David O. Johnson EECS 210 (Fall 2016) 2
Any Questions? Counting Techniques 3 David O. Johnson EECS 210 (Fall 2016)

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The Basics of Counting (Section 6.1) Basic Counting Principles The Product Rule The Sum Rule The Subtraction Rule The Division Rule Tree Diagrams Counting Techniques 4 David O. Johnson EECS 210 (Fall 2016)
Counting Combinatorics, the study of arrangements of objects, is an important part of discrete mathematics. This subject was studied as long ago as the seventeenth century, when combinatorial questions arose in the study of gambling games. Enumeration, the counting of objects with certain properties, is an important part of combinatorics. We must count objects to solve many different types of problems. For instance, counting is used to determine the complexity of algorithms. Counting is also required to determine whether there are enough telephone numbers or Internet protocol addresses to meet demand. Recently, it has played a key role in mathematical biology, especially in sequencing DNA. Furthermore, counting techniques are used extensively when probabilities of events are computed. First, we will study the basic rules of counting. The basic rules of counting can solve a tremendous variety of problems. For instance, we can use these rules to enumerate the different telephone numbers possible in the United States, the allowable passwords on a computer system, and the different orders in which the runners in a race can finish. Counting Techniques David O. Johnson EECS 210 (Fall 2016) 5

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Basic Rules of Counting Suppose that a password on a computer system consists of six, seven, or eight characters. Each of these characters must be a digit or a letter of the alphabet. Each password must contain at least one digit. How many such passwords are there? The techniques needed to answer this question and a wide variety of other counting problems will be introduced in this section. Counting problems arise throughout mathematics and computer science. We must count the successful outcomes of experiments and all the possible outcomes of these experiments to determine probabilities of discrete events. We need to count the number of operations used by an algorithm to study its time complexity. We will introduce the basic techniques of counting in this section. These methods serve as the foundation for almost all counting techniques. Counting Techniques David O. Johnson EECS 210 (Fall 2016) 6
Basic Counting Principles The Product Rule The Sum Rule The Subtraction Rule The Division Rule Counting Techniques David O. Johnson EECS 210 (Fall 2016) 7

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The Product Rule The Product Rule : A procedure can be broken down into a sequence of k tasks.
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