Chapter 11

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

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” © 2008 Pearson Addison-Wesley. All rights reserved

3. Chapter 1 Section 11-3 Using Permutations and Combinations © 2008 Pearson Addison-Wesley. All rights reserved

4. Using Permutations and Combinations • Permutations • Combinations • Guidelines on Which Method to Use © 2008 Pearson Addison-Wesley. All rights reserved

5. Permutations In the context of counting problems, arrangements are often called permutations; the number of permutations of n things taken r at a time is denoted nPr. Applying the fundamental counting principle to arrangements of this type gives nPr= n(n – 1)(n – 2)…[n – (r – 1)]. © 2008 Pearson Addison-Wesley. All rights reserved

6. Factorial Formula for Permutations The number of permutations, or arrangements, of n distinct things taken r at a time, where rn, can be calculated as © 2008 Pearson Addison-Wesley. All rights reserved

7. Example: Permutations Evaluate each permutation. a) 5P3 b) 6P6 Solution © 2008 Pearson Addison-Wesley. All rights reserved

8. Example: IDs How many ways can you select two letters followed by three digits for an ID if repeats are not allowed? Solution There are two parts: 1. Determine the set of two letters. 2. Determine the set of three digits. Part 1 Part 2 © 2008 Pearson Addison-Wesley. All rights reserved

9. Example: Building Numbers From a Set of Digits How many four-digit numbers can be written using the numbers from the set {1, 3, 5, 7, 9} if repetitions are not allowed? Solution Repetitions are not allowed and order is important, so we use permutations: © 2008 Pearson Addison-Wesley. All rights reserved

10. Combinations In the context of counting problems, subsets, where order of elements makes no difference, are often called combinations; the number of combinations of n things taken r at a time is denoted nCr. © 2008 Pearson Addison-Wesley. All rights reserved

11. Factorial Formula for Combinations The number of combinations, or subsets, of n distinct things taken r at a time, where rn, can be calculated as Note: © 2008 Pearson Addison-Wesley. All rights reserved

12. Example: Combinations Evaluate each combination. a) 5C3 b) 6C6 Solution © 2008 Pearson Addison-Wesley. All rights reserved

13. Example: Finding the Number of Subsets Find the number of different subsets of size 3 in the set {m, a, t, h, r, o, c, k, s}. Solution A subset of size 3 must have 3 distinct elements, so repetitions are not allowed. Order is not important. © 2008 Pearson Addison-Wesley. All rights reserved

14. Example: Finding the Number of Poker Hands A common form of poker involves hands (sets) of five cards each, dealt from a deck consisting of 52 different cards. How many different 5-card hands are possible? Solution Repetitions are not allowed and order is not important. © 2008 Pearson Addison-Wesley. All rights reserved

15. Guidelines on Which Method to Use © 2008 Pearson Addison-Wesley. All rights reserved