1. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there?

2. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime.

3. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny

4. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel

5. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel

6. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

7. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

8. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

9. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

10. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime There are 8 different arrangements of Heads and Tails.

11. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime There are 8 different arrangements of Heads and Tails. HHH HHT HTH HTT THH THT TTH TTT

12. MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime The structure that we used to count the number of outcomes is called a TREE DIAGRAM. …beginning with the penny …then the nickel …finally, the dime There are 8 different arrangements of Heads and Tails. HHH HHT HTH HTT THH THT TTH TTT

13. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting

14. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting FT There are 2 ways (T or F) to answer Question 1

15. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting FT If Q1 is answered T, Q2 can be answered either T or F. There are 2 ways (T or F) to answer Question 1

16. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT If Q1 is answered T, Q2 can be answered either T or F There are 2 ways (T or F) to answer Question 1

17. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT If Q1 is answered T, Q2 can be answered either T or F There are 2 ways (T or F) to answer Question 1

18. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT There are 2 ways (T or F) to answer Question 1 If Q1 is answered F, Q2 can be answered either T or F If Q1 is answered T, Q2 can be answered either T or F

19. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT There are 2 ways (T or F) to answer Question 1 If Q1 is answered F, Q2 can be answered either T or F TF If Q1 is answered T, Q2 can be answered either T or F

20. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT There are 2 ways (T or F) to answer Question 1 If Q1 is answered F, Q2 can be answered either T or F If Q1 is answered T, Q2 can be answered either T or F

21. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2

22. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Similarly, each possible answer to Q2 leads to 2 possible answers for Q3. Q3

23. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF

24. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF And each possible answer to Q3 leads to 2 possible answers for Q4. Q4

25. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T

26. If you are taking a 4 question T/F quiz: MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. How many different ways are there to answer all 4 questions?

27. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT

28. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF

29. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT

30. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF

31. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT

32. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF

33. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT

34. F If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF

35. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT F F

36. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF F F

37. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT F F

38. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF F F

39. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT F F

40. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT, FFTF F F

41. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT, FFTF, FFFT F F

42. If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T 123456789 10 11 12 13 14 15 16 There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT, FFTF, FFFT, FFFF} F F

43. MATH 110 Sec 12-1 Lecture: Intro to Counting

44. Any of the 5 letters could be selected first.

45. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Any of the 5 letters could be selected first.

46. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second.

47. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed.

48. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE

49. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE 5 ways

50. MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE A BCDE 5 ways A

51. MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE A BCDE 5 ways A

52. MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE If B is the first letter chosen, any of the 5 could still be chosen second. A BCDE 5 ways A

53. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If B is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed A BCDE 5 ways

54. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If B is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed A BCDE 5 ways …and because AB and BA are considered to be different.

55. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If B is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed A BCDE 5 ways …and because AB and BA are considered to be different. A BCDE 5 ways

56. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE 5 ways

57. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE 5 ways Similarly, if C is the first letter chosen, any of the 5 could still be chosen second.

58. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE 5 ways Similarly, if C is the first letter chosen, any of the 5 could still be chosen second. C A BCDE 5 ways

59. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE 5 ways

60. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE 5 ways The same logic holds for D and E being selected first.

61. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE 5 ways The same logic holds for D and E being selected first. A BCDE A BCDE 5 ways

62. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE A BCDE A BCDE 5 ways

63. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE A BCDE A BCDE So, there are 25 ways that this can be done. 5 ways

64. MATH 110 Sec 12-1 Lecture: Intro to Counting

65. Any of the 5 letters could be selected first.

66. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Any of the 5 letters could be selected first.

67. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the other 4 could still be chosen second.

68. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A can’t be used again because repetition is not allowed. If A is the first letter chosen, any of the other 4 could still be chosen second.

69. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE BCDE If A is the first letter chosen, any of the other 4 could still be chosen second. This is because repetition is not allowedA can’t be used again because repetition is not allowed.

70. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE BCDE 4 ways If A is the first letter chosen, any of the other 4 could still be chosen second. A can’t be used again because repetition is not allowed.

71. MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE BCDE 4 ways A

72. MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE If B is the first letter chosen, any of the other 4 could still be chosen second. A BCDE 4 ways

73. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE This is because repetition is not allowed BCDE 4 ways If B is the first letter chosen, any of the other 4 could still be chosen second.

74. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE …and because AB and BA are considered to be different. BCDE 4 ways If B is the first letter chosen, any of the other 4 could still be chosen second. This is because repetition is not allowed

75. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE …and because AB and BA are considered to be different. ACDEBCDE 4 ways If B is the first letter chosen, any of the other 4 could still be chosen second. This is because repetition is not allowed

76. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE ACDEBCDE 4 ways

77. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Similarly, if C is the first letter chosen, any of the other 4 could still be chosen second. ACDEBCDE 4 ways

78. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDEC ACDEBCDE 4 ways ABDE Similarly, if C is the first letter chosen, any of the other 4 could still be chosen second.

79. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE ACDEBCDE 4 ways ABDE

80. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. ACDEBCDE 4 ways ABDE

81. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. ACDEBCDE 4 ways ABDE ABCE ABCD

82. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE ACDEBCDE 4 ways ABDE ABCE ABCD

83. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE So, there are 20 ways that this can be done. ACDEBCDE 4 ways ABDE ABCE ABCD

84. MATH 110 Sec 12-1 Lecture: Intro to Counting

85. Any of the 5 letters could be selected first.

86. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Any of the 5 letters could be selected first.

87. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second.

88. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed.

89. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE

90. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE 5 ways

91. MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE A BCDE 5 ways A

92. MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second. A BCDE 5 ways A

93. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE This is because although repetition is allowed A BCDE 5 ways If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second.

94. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE 5 ways …AB and BA are NOT considered to be different. This is because although repetition is allowed If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second.

95. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE 5 ways BCDE This is because although repetition is allowed …AB and BA are NOT considered to be different. 4 ways If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second.

96. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE 5 ways BCDE 4 ways

97. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Similarly, if C is the first letter chosen, any of the 3 remaining letters (any but A or B) could still be chosen. second. A BCDE 5 ways BCDE 4 ways

98. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDEC CDE 3 ways A BCDE 5 ways BCDE 4 ways Similarly, if C is the first letter chosen, any of the 3 remaining letters (any but A or B) could still be chosen. second.

99. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE CDE 3 ways A BCDE 5 ways BCDE 4 ways

100. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. CDE 3 ways A BCDE 5 ways BCDE 4 ways

101. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. DEE 2 ways1 way CDE 3 ways A BCDE 5 ways BCDE 4 ways

102. MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE DEE 2 ways1 way CDE 3 ways A BCDE 5 ways BCDE 4 ways

103. MATH 110 Sec 12-1 Lecture: Intro to Counting So, there are 15 ways that this can be done. A BCDE DEE 2 ways1 way CDE 3 ways A BCDE 5 ways BCDE 4 ways

104. MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible?

105. MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else.

106. MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else.

107. MATH 110 Sec 12-1 Lecture: Intro to Counting 112121313141415151616 212122323242425252626 313132323343435353636 414142424343445454646 515152525353545455656 616162626363646465656 Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else.

108. MATH 110 Sec 12-1 Lecture: Intro to Counting 112121313141415151616 212122323242425252626 313132323343435353636 414142424343445454646 515152525353545455656 616162626363646465656 Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else. There are 6 x 6 = 36 entries in this table.

109. MATH 110 Sec 12-1 Lecture: Intro to Counting 112121313141415151616 212122323242425252626 313132323343435353636 414142424343445454646 515152525353545455656 616162626363646465656 Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else. Each entry in this table represents a different roll so there are 36 different possible rolls. There are 6 x 6 = 36 entries in this table.

110. MATH 110 Sec 12-1 Lecture: Intro to Counting 112121313141415151616 212122323242425252626 313132323343435353636 414142424343445454646 515152525353545455656 616162626363646465656 Suppose you roll 2 dice. The first is red & the second is green. In how many ways can you roll a sum of 9?

111. MATH 110 Sec 12-1 Lecture: Intro to Counting 112121313141415151616 212122323242425252626 313132323343435353636 414142424343445454646 515152525353545455656 616162626363646465656 Suppose you roll 2 dice. The first is red & the second is green. In how many ways can you roll a sum of 9?

112. MATH 110 Sec 12-1 Lecture: Intro to Counting 112121313141415151616 212122323242425252626 313132323343435353636 414142424343445454646 515152525353545455656 616162626363646465656 Suppose you roll 2 dice. The first is red & the second is green. In how many ways can you roll a sum of 9? So there are 4 ways to roll a sum of 9.