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

2. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-2 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 11-3-3 Chapter 1 Section 11-3 Using Permutations and Combinations

4. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-4 Using Permutations and Combinations Permutations Combinations Guidelines on Which Method to Use

5. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-5 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 n P r. Applying the fundamental counting principle to arrangements of this type gives n P r = n(n – 1)(n – 2)…[n – (r – 1)].

6. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-6 Factorial Formula for Permutations The number of permutations, or arrangements, of n distinct things taken r at a time, where r n, can be calculated as

7. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-7 Example: Permutations Evaluate each permutation. a) 5 P 3 b) 6 P 6 Solution

8. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-8 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

9. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-9 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:

10. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-10 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 n C r.

11. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-11 Factorial Formula for Combinations The number of combinations, or subsets, of n distinct things taken r at a time, where r n, can be calculated as Note:

12. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-12 Example: Combinations Evaluate each combination. a) 5 C 3 b) 6 C 6 Solution

13. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-13 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.

14. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-14 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.

15. © 2008 Pearson Addison-Wesley. All rights reserved 11-3-15 Guidelines on Which Method to Use PermutationsCombinations Number of ways of selecting r items out of n items Repetitions are not allowed Order is important.Order is not important. Arrangements of n items taken r at a time Subsets of n items taken r at a time n P r = n!/(n – r)! n C r = n!/[ r!(n – r)!] Clue words: arrangement, schedule, order Clue words: group, sample, selection