1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 9 Discrete Mathematics

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.1 Basic Combinatorics

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 6 What you’ll learn about Discrete Versus Continuous The Importance of Counting The Multiplication Principle of Counting Permutations Combinations Subsets of an n-Set … and why Counting large sets is easy if you know the correct formula.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 7 Continuous versus Discrete Continuous mathematics – field dealing with systems of contiguous, dimensionless points, each a measurable distance from some reference point In continuous systems, there are an infinite number of points in any interval. Discrete mathematics – field dealing with systems of discrete units that can be counted, e.g. the letters in the alphabet, peanuts in a jar, etc. In discrete systems, there are a finite number of objects in any set, such that the objects may be counted and the total number of the objects ascertained.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 8 Example Counting the Number of Ways a Procedure May Be Done How many different ways can four books be arranged on a book shelf?

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9 Multiplication Principle of Counting

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 10 Example Using the Multiplication Principle If six girls and five boys are selected to the homecoming court, how many possible king-queen pairs are there?

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 11 Permutations of an n-Set A permutation is a result of an experiment in which the order of the members of the set is important. If a set has n members, it is called an n-set There are n! possible permutations of an n-set.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 12 Example Distinguishable Permutations Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 13 Distinguishable Permutations

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 14 Example Permutations of an n-set Containing Identical Multiples How many different arrangements of 12 M & Ms may be made, if there are three reds, two blues, five browns, and two yellows?

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 15 Permutations Counting Formula

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 16 Example Permutations of an n-set Taken r at a Time How many possible combinations of ASB officers are possible from a class of 325?

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 17 Combination Counting Formula

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 18 Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 19 Formula for Counting Subsets of an n-Set

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 20 Homework Homework Assignment #27 Review Section 9.1 Page 708, Exercises: 1 – 45 (EOO)

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.2 The Binomial Theorem

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 22 Quick Review

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 23 Quick Review Solutions

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 24 What you’ll learn about Powers of Binomials Pascal’s Triangle The Binomial Theorem Factorial Identities … and why The Binomial Theorem is a marvelous study in combinatorial patterns.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 25 Binomial Coefficient

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 26 Example Using n C r to Expand a Binomial

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 27 Pascal’s Triangle Pascal’s Triangle is a listing of the coefficients of the terms in the expansion of a binomial.

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 28 Example Using Pascal’s Triangle to Expand a Binomial

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 29 The Binomial Theorem

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 30 Expanding a Binomial Points to note about Binomial Expansions: 1.The number of terms in the expansion is one more than the exponent. 2.The sum of the exponents of the variables in a term is always the exponent to which the binomial is raised, assuming both variables are first order in the initial expression. 3.The coefficient of the second term is the same as the exponent.

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 31 Example Evaluating a Coefficient in a Binomial Expansion by hand

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 32 Example Writing the Specified Term of a Binomial Expansion

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 33 Basic Factorial Identities