ch3p1 - SCSI1013 Discrete Structures CHAPTER 3 COUNTING...

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CHAPTER 3 COUNTING METHODS [Part 1] SCSI1013: Discrete Structures 2014/2015 – Sem. 1
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Basic Counting Principles Counting principle is all about choices we might make given many possibilities. It is used to find the number of possible outcomes. It provides a basis of computing probabilities of discrete events. 2
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Some sample of counting problems: Problem 1 - How many ways are there to seat n couples at a round table, such that each couple sits together? Problem 2- How many ways are there to express a positive integer n as a sum of positive integers? 3 Basic Counting Principles
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Problem 3 - There are three boxes containing books. The first box contains 15 mathematics books by different authors, the second box contains 12 chemistry books by different authors, and the third box contains 10 computer science books by different authors. A student wants to take a book from one of the three boxes. In how many ways can the student do this? 4 Basic Counting Principles
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Problem 4 - The department will award a free computer to either a CS student or a CS professor. How many different choices are there, if there are 530 students and 15 professors? Basic Counting Principles
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Problem 5 - A scholarship is available, and the student to receive this scholarship must be chosen from the Mathematics, Computer Science, or the Engineering Department. How many different choices are there for this student if there are 38 qualified students from the Mathematics Department, 45 qualified students from the Computer Science Department and 27 qualified students from the Engineering Department?
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There are a number of basic principles that we can use to solve such problems: 1) Addition Principle 2) Multiplication Principle 7 Basic Counting Principles
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8 Addition Principle Suppose that tasks T 1 , T 2 ,… T k can be done in n 1 , n 2 ,… n k ways, respectively. If all these tasks are independent of each other, then the number of ways to do one of these tasks is n 1 + n 2 +….+ n k
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If a task can be done in n 1 ways and a second task in n 2 ways, and if these two tasks cannot be done at the same time, then there are n 1 + n 2 ways to do either task. 9 Basic Counting Principles
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10 Example 1 We want to find the number of integers between 5 and 50 that end with 1 or 7. Solution: Let T denote this task. We divide T into the following tasks. T 1 : find all integers between 5 and 50 that end with 1. T 2 : find all integers between 5 and 50 that end with 7.
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11 T 1 : find all integers between 5 and 50 that end with 1. 11, 21, 31, 41 => 4 ways T 2 : find all integers between 5 and 50 that end with 7.
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