1. Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved

2. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-2 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”

3. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-3 Chapter 1 Section 11-2 Using the Fundamental Counting Principle

4. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-4 Using the Fundamental Counting Principle Uniformity and the Fundamental Counting Principle Factorials Arrangements of Objects

5. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-5 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.

6. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-6 Fundamental Counting Principle When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n 1 ways, the second part can be done in n 2 ways, and so on through the k th part, which can be done in n k ways, then the total number of ways to complete the task is given by the product

7. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-7 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 Part of TaskSelect first digitSelect second digit Number of ways5 (0 can’t be used) 6 There are 5(6) = 30 two-digit numbers.

8. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-8 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 Part of Task Select first digit Select second digit Number of ways 55 (repeated digits not allowed) There are 5(5) = 25 two-digit numbers.

9. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-9 Example: Two-Digit Numbers with Restrictions How many ways can you select two letters followed by three digits for an ID? Solution Part of Task First letter Second letter Digit Number of ways 26 10 There are 26(26)(10)(10)(10) = 676,000 IDs possible.

10. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-10 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!.

11. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-11 Factorial Formula For any counting number n, the quantity n factorial is given by

12. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-12 Example: Evaluate each expression. a) 4!b) (4 – 1)!c) Solution

13. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-13 Definition of Zero Factorial

14. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-14 Arrangements of Objects When finding the total number of ways to arrange a given number of distinct objects, we can use a factorial.

15. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-15 Arrangements of n Distinct Objects The total number of different ways to arrange n distinct objects is n!.

16. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-16 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.

17. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-17 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 n 1 are of one kind, n 2 are of another kind, …, and n k are of yet another kind), is given by

18. © 2008 Pearson Addison-Wesley. All rights reserved 11-2-18 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