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## Chapter 11

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**Chapter 11**Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved**Chapter 11: Counting Methods**11.1 Counting by Systematic Listing 11.2 Using the Fundamental Counting Principle 11.3 Using Permutations and Combinations 11.4 Using Pascal’s Triangle 11.5 Counting Problems Involving “Not” and “Or” © 2008 Pearson Addison-Wesley. All rights reserved**Chapter 1**Section 11-2 Using the Fundamental Counting Principle © 2008 Pearson Addison-Wesley. All rights reserved**Using the Fundamental Counting Principle**• Uniformity and the Fundamental Counting Principle • Factorials • Arrangements of Objects © 2008 Pearson Addison-Wesley. All rights reserved**Uniformity Criterion for Multiple-Part Tasks**A multiple-part task is said to satisfy the uniformity criterion if the number of choices for any particular part is the same no matter which choices were selected for the previous parts. © 2008 Pearson Addison-Wesley. All rights reserved**Fundamental Counting Principle**When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n1 ways, the second part can be done in n2 ways, and so on through the kth part, which can be done in nk ways, then the total number of ways to complete the task is given by the product © 2008 Pearson Addison-Wesley. All rights reserved**Example: Two-Digit Numbers**How many two-digit numbers can be made from the set {0, 1, 2, 3, 4, 5}? (numbers can’t start with 0.) Solution There are 5(6) = 30 two-digit numbers. © 2008 Pearson Addison-Wesley. All rights reserved**Example: Two-Digit Numbers with Restrictions**How many two-digit numbers that do not contain repeated digits can be made from the set {0, 1, 2, 3, 4, 5} ? Solution There are 5(5) = 25 two-digit numbers. © 2008 Pearson Addison-Wesley. All rights reserved**Example: Two-Digit Numbers with Restrictions**How many ways can you select two letters followed by three digits for an ID? Solution There are 26(26)(10)(10)(10) = 676,000 IDs possible. © 2008 Pearson Addison-Wesley. All rights reserved**Factorials**For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!. © 2008 Pearson Addison-Wesley. All rights reserved**Factorial Formula**For any counting number n, the quantity n factorial is given by © 2008 Pearson Addison-Wesley. All rights reserved**Example:**Evaluate each expression. a) 4! b) (4 – 1)! c) Solution © 2008 Pearson Addison-Wesley. All rights reserved**Definition of Zero Factorial**© 2008 Pearson Addison-Wesley. All rights reserved**Arrangements of Objects**When finding the total number of ways to arrange a given number of distinct objects, we can use a factorial. © 2008 Pearson Addison-Wesley. All rights reserved**Arrangements of n Distinct Objects**The total number of different ways to arrange n distinct objects is n!. © 2008 Pearson Addison-Wesley. All rights reserved**Example: Arranging Books**How many ways can you line up 6 different books on a shelf? Solution The number of ways to arrange 6 distinct objects is 6! = 720. © 2008 Pearson Addison-Wesley. All rights reserved**Arrangements of n Objects Containing Look-Alikes**The number of distinguishable arrangements of n objects, where one or more subsets consist of look-alikes (say n1 are of one kind, n2 are of another kind, …, and nk are of yet another kind), is given by © 2008 Pearson Addison-Wesley. All rights reserved**Example: Distinguishable Arrangements**Determine the number of distinguishable arrangements of the letters of the word INITIALLY. Solution 9 letters total 3 I’s and 2 L’s © 2008 Pearson Addison-Wesley. All rights reserved