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 Multiplication Principle of Counting

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 8 Example Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9 Example Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. You can fill in the first blank 26 ways, the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways, the sixth blank 10 ways, and the seventh blank 10 ways. By the Multiplication Principle, there are 26×26×26×26×10×10×10 = 456,976,000 possible license plates.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 10 Permutations of an n-Set There are n! permutations of an n-set.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 11 Example Distinguishable Permutations Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.

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. Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.

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 Permutations Counting Formula

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

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 16 Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 17 Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 18 Formula for Counting Subsets of an n-Set

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

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 20 Quick Review

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 21 Quick Review Solutions

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

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 23 Binomial Coefficient

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 24 Example Using n C r to Expand a Binomial

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 25 Example Using n C r to Expand a Binomial

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 26 Recursion Formula for Pascal’s Triangle

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 27 The Binomial Theorem

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 28 Basic Factorial Identities

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.3 Probability

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 30 Quick Review

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 31 Quick Review Solutions

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 32 What you’ll learn about Sample Spaces and Probability Functions Determining Probabilities Venn Diagrams and Tree Diagrams Conditional Probability Binomial Distributions … and why Everyone should know how mathematical the “laws of chance” really are.

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 33 Probability of an Event (Equally Likely Outcomes)

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 34 Probability Distribution for the Sum of Two Fair Dice OutcomeProbability 21/36 32/36 43/36 54/36 65/36 76/36 85/36 94/36 103/36 112/36 121/36

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 35 Example Rolling the Dice Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 36 Example Rolling the Dice Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 37 Probability Function

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 38 Probability of an Event (Outcomes not Equally Likely)

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 39 Strategy for Determining Probabilities

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 40 Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 41 Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 42 Multiplication Principle of Probability Suppose an event A has probability p 1 and an event B has probability p 2 under the assumption that A occurs. Then the probability that both A and B occur is p 1 p 2.

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 43 Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 44 Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 45 Conditional Probability Formula

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 46 Binomial Distribution

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 47 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 48 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 49 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 50 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.4 Sequences

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 52 Quick Review

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 53 Quick Review Solutions

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 54 What you’ll learn about Infinite Sequences Limits of Infinite Sequences Arithmetic and Geometric Sequences Sequences and Graphing Calculators … and why Infinite sequences, especially those with finite limits, are involved in some key concepts of calculus.

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 55 Limit of a Sequence

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 56 Example Finding Limits of Sequences

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 57 Example Finding Limits of Sequences

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 58 Arithmetic Sequence

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 59 Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. -2, 1, 4, 7, …

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 60 Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. -2, 1, 4, 7, …

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 61 Geometric Sequence

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 62 Example Defining Geometric Sequences Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. 2, 6, 18,…

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 63 Example Defining Geometric Sequences Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. 2, 6, 18,…

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 64 Sequences and Graphing Calculators One way to graph a explicitly defined sequences is as scatter plots of the points of the form (k,a k ). A second way is to use the sequence mode on a graphing calculator.

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 65 The Fibonacci Sequence

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.5 Series

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 67 Quick Review

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 68 Quick Review Solutions

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 69 What you’ll learn about Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergences of Geometric Series … and why Infinite series are at the heart of integral calculus.

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 70 Summation Notation

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 71 Sum of a Finite Arithmetic Sequence

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 72 Example Summing the Terms of an Arithmetic Sequence

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 73 Example Summing the Terms of an Arithmetic Sequence

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 74 Sum of a Finite Geometric Sequence

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 75 Infinite Series

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 76 Sum of an Infinite Geometric Series

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 77 Example Summing Infinite Geometric Series

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 78 Example Summing Infinite Geometric Series

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.6 Mathematical Induction

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 80 Quick Review

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 81 Quick Review Solutions

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 82 What you’ll learn about The Tower of Hanoi Problem Principle of Mathematical Induction Induction and Deduction … and why The principle of mathematical induction is a valuable technique for proving combinatorial formulas.

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 83 The Tower of Hanoi Solution The minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2 n – 1.

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 84 Principle of Mathematical Induction Let P n be a statement about the integer n. Then P n is true for all positive integers n provided the following conditions are satisfied: 1. (the anchor) P 1 is true; 2. (inductive step) if P k is true, then P k+1 is true.

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.7 Statistics and Data (Graphical)

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 86 Quick Review

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 87 Quick Review Solutions

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 88 What you’ll learn about Statistics Displaying Categorical Data Stemplots Frequency Tables Histograms Time Plots … and why Graphical displays of data are increasingly prevalent in professional and popular media. We all need to understand them.

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 89 Leading Causes of Death in the United States in 2001 Cause of DeathNumber of DeathsPercentage Heart Disease700,14229.0 Cancer553,76822.9 Stroke163,5386.8 Other1,018,97741.3 Source: National Center for Health Statistics, as reported in The World Almanac and Book of Facts 2005.

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 90 Bar Chart, Pie Chart, Circle Graph

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 91 Example Making a Stemplot Make a stemplot for the given data. 12.3 23.4 12.0 24.5 23.7 18.7 22.4 19.5 24.5 24.6

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 92 Example Making a Stemplot Make a stemplot for the given data. 12.3 23.4 12.0 24.5 23.7 18.7 22.4 19.5 24.5 24.6 StemLeaf 120,3 187 195 224 234,7 245,5,6

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 93 Time Plot

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.8 Statistics and Data (Algebraic)

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 95 Quick Review

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 96 Quick Review Solutions

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 97 What you’ll learn about Parameters and Statistics Mean, Median, and Mode The Five-Number Summary Boxplots Variance and Standard Deviation Normal Distributions … and why The language of statistics is becoming more commonplace in our everyday world.

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 98 Mean

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 99 Median The median of a list of n numbers {x 1,x 2,…,x n } arranged in order (either ascending or descending) is the middle number if n is odd, and the mean of the two middle numbers if n is even.

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 100 Mode The mode of a list of numbers is the number that appears most frequently in the list.

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 101 Example Finding Mean, Median, and Mode Find the (a) mean, (b) median, and (c) mode of the data: 3, 6, 5, 7, 8, 10, 6, 2, 4, 6

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 102 Example Finding Mean, Median, and Mode Find the (a) mean, (b) median, and (c) mode of the data: 3, 6, 5, 7, 8, 10, 6, 2, 4, 6

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 103 Weighted Mean

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 104 Five-Number Summary

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 105 Boxplot

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 106 Outlier A number in a data set can be considered an outlier if it is more than 1.5×IQR below the first quartile or above the third quartile.

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 107 Standard Deviation

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 108 Normal Curve

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 109 The 68-95-99.7 Rule If the data for a population are normally distributed with mean μ and standard deviation σ, then Approximately 68% of the data lie between μ - 1σ and μ + 1σ. Approximately 95% of the data lie between μ - 2σ and μ + 2σ. Approximately 99.7% of the data lie between μ - 3σ and μ + 3σ.

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 110 The 68-95-99.7 Rule

111. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 111 Chapter Test

112. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 112 Chapter Test

113. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 113 Chapter Test

114. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 114 Chapter Test Solutions

115. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 115 Chapter Test Solutions

116. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 116 Chapter Test Solutions