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Math 1300: Section 7- 3 Basic Counting Principles

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Math 1300: Section 7- 3 Basic Counting Principles

1. 1. Addition Principle Venn Diagrams Multiplication PrincipleMath 1300 Finite MathematicsSection 7-3: Basic Counting Principles Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
2. 2. Addition Principle Venn Diagrams Multiplication PrincipleExample: If enrollment in a section of Math 1300 consists of 13males and 15 females, then it is clear that there are a total of 28students in the class. university-logo Jason Aubrey Math 1300 Finite Mathematics
3. 3. Addition Principle Venn Diagrams Multiplication PrincipleExample: If enrollment in a section of Math 1300 consists of 13males and 15 females, then it is clear that there are a total of 28students in the class.To represent this in terms of set operations, we would ﬁrstassign names to the sets. Let M = set of male students in the section F = set of female students in the section university-logo Jason Aubrey Math 1300 Finite Mathematics
4. 4. Addition Principle Venn Diagrams Multiplication PrincipleExample: If enrollment in a section of Math 1300 consists of 13males and 15 females, then it is clear that there are a total of 28students in the class.To represent this in terms of set operations, we would ﬁrstassign names to the sets. Let M = set of male students in the section F = set of female students in the sectionNotice that M ∪ F is the set of all students in the class, and thatM ∩ F = ∅. university-logo Jason Aubrey Math 1300 Finite Mathematics
5. 5. Addition Principle Venn Diagrams Multiplication PrincipleExample: If enrollment in a section of Math 1300 consists of 13males and 15 females, then it is clear that there are a total of 28students in the class.To represent this in terms of set operations, we would ﬁrstassign names to the sets. Let M = set of male students in the section F = set of female students in the sectionNotice that M ∪ F is the set of all students in the class, and thatM ∩ F = ∅. The total number of students in the class is thenrepresented by n(M ∪ F ), and we have n(M ∪ F ) = n(M) + n(F ) = 13 + 15 = 28. university-logo Jason Aubrey Math 1300 Finite Mathematics
6. 6. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose we are told that an economics classconsists of 22 business majors and 16 journalism majors. university-logo Jason Aubrey Math 1300 Finite Mathematics
7. 7. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose we are told that an economics classconsists of 22 business majors and 16 journalism majors.Let B represent the set of business majors and J represent theset of journalism majors. university-logo Jason Aubrey Math 1300 Finite Mathematics
8. 8. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose we are told that an economics classconsists of 22 business majors and 16 journalism majors.Let B represent the set of business majors and J represent theset of journalism majors.Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of allstudents in the class. university-logo Jason Aubrey Math 1300 Finite Mathematics
9. 9. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose we are told that an economics classconsists of 22 business majors and 16 journalism majors.Let B represent the set of business majors and J represent theset of journalism majors.Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of allstudents in the class.Can we conclude that n(B ∪ J) = 22 + 16 = 38? university-logo Jason Aubrey Math 1300 Finite Mathematics
10. 10. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose we are told that an economics classconsists of 22 business majors and 16 journalism majors.Let B represent the set of business majors and J represent theset of journalism majors.Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of allstudents in the class.Can we conclude that n(B ∪ J) = 22 + 16 = 38?No! This would double count double majors. university-logo Jason Aubrey Math 1300 Finite Mathematics
11. 11. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose we are told that an economics classconsists of 22 business majors and 16 journalism majors.Let B represent the set of business majors and J represent theset of journalism majors.Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of allstudents in the class.Can we conclude that n(B ∪ J) = 22 + 16 = 38?No! This would double count double majors.The set B ∩ J represents the set of students majoring in bothbusiness and journalism. If B ∩ J = ∅, then we must avoidcounting these students twice. university-logo Jason Aubrey Math 1300 Finite Mathematics
12. 12. Addition Principle Venn Diagrams Multiplication PrincipleIf we had, say, 7 double majors in the class, then n(B ∩ J) = 7 university-logo Jason Aubrey Math 1300 Finite Mathematics
13. 13. Addition Principle Venn Diagrams Multiplication PrincipleIf we had, say, 7 double majors in the class, then n(B ∩ J) = 7And the correct count would be n(B ∪ J) = n(B) + n(J) − n(B ∩ J) = 22 + 16 − 7 = 31 university-logo Jason Aubrey Math 1300 Finite Mathematics
14. 14. Addition Principle Venn Diagrams Multiplication PrincipleTheorem (Addition Principle (For Counting))For any two sets A and B, n(A ∪ B) = n(A) + n(B) − n(A ∩ B)If A and B are disjoint, then n(A ∪ B) = n(A) + n(B) university-logo Jason Aubrey Math 1300 Finite Mathematics
15. 15. Addition Principle Venn Diagrams Multiplication PrincipleExample: A marketing survey of a group of kids indicated that25 owned a Nintendo DS and 15 owned a Wii. If 10 kids hadboth a DS and a Wii, how many kids interviewed have a DS ora Wii? university-logo Jason Aubrey Math 1300 Finite Mathematics
16. 16. Addition Principle Venn Diagrams Multiplication PrincipleExample: A marketing survey of a group of kids indicated that25 owned a Nintendo DS and 15 owned a Wii. If 10 kids hadboth a DS and a Wii, how many kids interviewed have a DS ora Wii?Let D represent the set of kids with a DS, and let W representthe set of kids with a Wii. university-logo Jason Aubrey Math 1300 Finite Mathematics
17. 17. Addition Principle Venn Diagrams Multiplication PrincipleExample: A marketing survey of a group of kids indicated that25 owned a Nintendo DS and 15 owned a Wii. If 10 kids hadboth a DS and a Wii, how many kids interviewed have a DS ora Wii?Let D represent the set of kids with a DS, and let W representthe set of kids with a Wii. n(D ∪ W ) = n(D) + n(W ) − n(D ∩ W ) university-logo Jason Aubrey Math 1300 Finite Mathematics
18. 18. Addition Principle Venn Diagrams Multiplication PrincipleExample: A marketing survey of a group of kids indicated that25 owned a Nintendo DS and 15 owned a Wii. If 10 kids hadboth a DS and a Wii, how many kids interviewed have a DS ora Wii?Let D represent the set of kids with a DS, and let W representthe set of kids with a Wii. n(D ∪ W ) = n(D) + n(W ) − n(D ∩ W ) n(D ∪ W ) = 25 + 15 − 10 = 30 university-logo Jason Aubrey Math 1300 Finite Mathematics
19. 19. Addition Principle Venn Diagrams Multiplication PrincipleExample: A marketing survey of a group of kids indicated that25 owned a Nintendo DS and 15 owned a Wii. If 10 kids hadboth a DS and a Wii, how many kids interviewed have a DS ora Wii?Let D represent the set of kids with a DS, and let W representthe set of kids with a Wii. n(D ∪ W ) = n(D) + n(W ) − n(D ∩ W ) n(D ∪ W ) = 25 + 15 − 10 = 30Number of kids with a DS or a Wii: 30. university-logo Jason Aubrey Math 1300 Finite Mathematics
20. 20. Addition Principle Venn Diagrams Multiplication PrincipleIn problems which involve more than two sets or which involvecomplements of sets, it is often helpful to draw a VennDiagram. university-logo Jason Aubrey Math 1300 Finite Mathematics
21. 21. Addition Principle Venn Diagrams Multiplication PrincipleIn problems which involve more than two sets or which involvecomplements of sets, it is often helpful to draw a VennDiagram.Example: In a certain class, there are 23 majors in Psychology,16 majors in English and 7 students who are majoring in bothPsychology and English. If there are 50 students in the class,how many students are majoring in neither of these subjects?How many students are majoring in Psychology alone? university-logo Jason Aubrey Math 1300 Finite Mathematics
22. 22. Addition Principle Venn Diagrams Multiplication PrincipleIn problems which involve more than two sets or which involvecomplements of sets, it is often helpful to draw a VennDiagram.Example: In a certain class, there are 23 majors in Psychology,16 majors in English and 7 students who are majoring in bothPsychology and English. If there are 50 students in the class,how many students are majoring in neither of these subjects?How many students are majoring in Psychology alone?Let P represent the set of Psychology majors and let Erepresent the set of English majors. university-logo Jason Aubrey Math 1300 Finite Mathematics
23. 23. Addition Principle Venn Diagrams Multiplication PrincipleThen n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are alsogiven that there are 50 students in the class, so n(U) = 50.Now we draw a Venn Diagram: university-logo Jason Aubrey Math 1300 Finite Mathematics
24. 24. Addition Principle Venn Diagrams Multiplication PrincipleThen n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are alsogiven that there are 50 students in the class, so n(U) = 50.Now we draw a Venn Diagram: n(U) = 50 P E university-logo Jason Aubrey Math 1300 Finite Mathematics
25. 25. Addition Principle Venn Diagrams Multiplication PrincipleThen n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are alsogiven that there are 50 students in the class, so n(U) = 50.Now we draw a Venn Diagram: n(U) = 50 P E 7 university-logo Jason Aubrey Math 1300 Finite Mathematics
26. 26. Addition Principle Venn Diagrams Multiplication PrincipleThen n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are alsogiven that there are 50 students in the class, so n(U) = 50.Now we draw a Venn Diagram: n(U) = 50 P E 16 7 university-logo Jason Aubrey Math 1300 Finite Mathematics
27. 27. Addition Principle Venn Diagrams Multiplication PrincipleThen n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are alsogiven that there are 50 students in the class, so n(U) = 50.Now we draw a Venn Diagram: n(U) = 50 P E 16 7 9 university-logo Jason Aubrey Math 1300 Finite Mathematics
28. 28. Addition Principle Venn Diagrams Multiplication PrincipleThen n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are alsogiven that there are 50 students in the class, so n(U) = 50.Now we draw a Venn Diagram: n(U) = 50 P E 16 7 9 18 university-logo Jason Aubrey Math 1300 Finite Mathematics
29. 29. Addition Principle Venn Diagrams Multiplication PrincipleExample: A survey of 100 college faculty who exerciseregularly found that 45 jog, 30 swim, 20 cycle, 6 jog and swim,1 jogs and cycles, 5 swim and cycle, and 1 does all three. Howmany of the faculty members do not do any of these threeactivities? How many just jog? university-logo Jason Aubrey Math 1300 Finite Mathematics
30. 30. Addition Principle Venn Diagrams Multiplication PrincipleLet A and B be sets and A ⊂ U, B ⊂ U, university-logo Jason Aubrey Math 1300 Finite Mathematics
31. 31. Addition Principle Venn Diagrams Multiplication PrincipleLet A and B be sets and A ⊂ U, B ⊂ U, n(A ) = n(U) − n(A) university-logo Jason Aubrey Math 1300 Finite Mathematics
32. 32. Addition Principle Venn Diagrams Multiplication PrincipleLet A and B be sets and A ⊂ U, B ⊂ U, n(A ) = n(U) − n(A) DeMorgan’s Laws university-logo Jason Aubrey Math 1300 Finite Mathematics
33. 33. Addition Principle Venn Diagrams Multiplication PrincipleLet A and B be sets and A ⊂ U, B ⊂ U, n(A ) = n(U) − n(A) DeMorgan’s Laws (A ∪ B) = A ∩ B (A ∩ B) = A ∪ B university-logo Jason Aubrey Math 1300 Finite Mathematics
34. 34. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U A B university-logo Jason Aubrey Math 1300 Finite Mathematics
35. 35. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U A B63 university-logo Jason Aubrey Math 1300 Finite Mathematics
36. 36. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 10863 university-logo Jason Aubrey Math 1300 Finite Mathematics
37. 37. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 n(A ∩ B) = 180 − n(A ∪ B ) = 180 − 108 = 7263 university-logo Jason Aubrey Math 1300 Finite Mathematics
38. 38. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 n(A ∩ B) = 180 − n(A ∪ B ) = 180 − 108 = 72 n(A) = 180 − n(A ) = 9963 university-logo Jason Aubrey Math 1300 Finite Mathematics
39. 39. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 n(A ∩ B) = 180 − n(A ∪ B ) = 180 − 108 = 72 n(A) = 180 − n(A ) = 9963 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
40. 40. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 n(A ∩ B) = 180 − n(A ∪ B ) 72 = 180 − 108 = 72 n(A) = 180 − n(A ) = 9963 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
41. 41. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 n(A ∩ B) = 180 − n(A ∪ B ) 72 = 180 − 108 = 72 n(A) = 180 − n(A ) = 9963 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
42. 42. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 n(A ∩ B) = 180 − n(A ∪ B ) 27 72 = 180 − 108 = 72 n(A) = 180 − n(A ) = 9963 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
43. 43. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose that A and B are sets with n(A ) = 81,n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine thenumber of elements in each of the four disjoint subsets in thefollowing Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 n(A ∩ B) = 180 − n(A ∪ B ) 27 72 18 = 180 − 108 = 72 n(A) = 180 − n(A ) = 9963 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
44. 44. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose you have 4 pairs of pants in your closet, 3different shirts and 2 pairs of shoes. How many different wayscan you choose an outﬁt consisting of one pair of pants, oneshirt and one pair of shoes? university-logo Jason Aubrey Math 1300 Finite Mathematics
45. 45. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose you have 4 pairs of pants in your closet, 3different shirts and 2 pairs of shoes. How many different wayscan you choose an outﬁt consisting of one pair of pants, oneshirt and one pair of shoes?Let’s consider this as a sequence of operations: university-logo Jason Aubrey Math 1300 Finite Mathematics
46. 46. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose you have 4 pairs of pants in your closet, 3different shirts and 2 pairs of shoes. How many different wayscan you choose an outﬁt consisting of one pair of pants, oneshirt and one pair of shoes?Let’s consider this as a sequence of operations: O1 Choose a pair of pants university-logo Jason Aubrey Math 1300 Finite Mathematics
47. 47. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose you have 4 pairs of pants in your closet, 3different shirts and 2 pairs of shoes. How many different wayscan you choose an outﬁt consisting of one pair of pants, oneshirt and one pair of shoes?Let’s consider this as a sequence of operations: O1 Choose a pair of pants O2 Choose a shirt university-logo Jason Aubrey Math 1300 Finite Mathematics
48. 48. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose you have 4 pairs of pants in your closet, 3different shirts and 2 pairs of shoes. How many different wayscan you choose an outﬁt consisting of one pair of pants, oneshirt and one pair of shoes?Let’s consider this as a sequence of operations: O1 Choose a pair of pants O2 Choose a shirt O3 Choose a pair of shoes university-logo Jason Aubrey Math 1300 Finite Mathematics
49. 49. Addition Principle Venn Diagrams Multiplication PrincipleExample: Suppose you have 4 pairs of pants in your closet, 3different shirts and 2 pairs of shoes. How many different wayscan you choose an outﬁt consisting of one pair of pants, oneshirt and one pair of shoes?Let’s consider this as a sequence of operations: O1 Choose a pair of pants O2 Choose a shirt O3 Choose a pair of shoesNow for each operation, there is a speciﬁed number of ways toperform this operation: Operation Number of Ways O1 N1 = 4 O2 N2 = 3 O3 N3 = 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
50. 50. Addition Principle Venn Diagrams Multiplication PrincipleSo we have i Oi Ni 1 O1 4 2 O2 3 3 O3 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
51. 51. Addition Principle Venn Diagrams Multiplication PrincipleSo we have i Oi Ni 1 O1 4 2 O2 3 3 O3 2Then we can draw a tree diagram to see that there areN1 · N2 · N3 = 4(3)(2) = 24 different outﬁts. university-logo Jason Aubrey Math 1300 Finite Mathematics
52. 52. Addition Principle Venn Diagrams Multiplication PrincipleTheorem (Multiplication Principle)If two operations O1 and O2 are performed in order, with N1possible outcomes for the ﬁrst operation and N2 possibleoutcomes for the second operation, then there are N1 · N2possible combined outcomes for the ﬁrst operation followed bythe second. university-logo Jason Aubrey Math 1300 Finite Mathematics
53. 53. Addition Principle Venn Diagrams Multiplication PrincipleTheorem (Generalized Multiplication Principle)In general, if n operations O1 , O2 , · · · , On are performed inorder, with possible number of outcomes N1 , N2 , . . . , Nn ,respectively, then there are N1 · N2 · · · Nnpossible combined outcomes of the operations performed in thegiven order. university-logo Jason Aubrey Math 1300 Finite Mathematics
54. 54. Addition Principle Venn Diagrams Multiplication PrincipleExample: A fair 6-sided die is rolled 5 times, and each time theresulting sequence of 5 numbers is recorded. university-logo Jason Aubrey Math 1300 Finite Mathematics
55. 55. Addition Principle Venn Diagrams Multiplication PrincipleExample: A fair 6-sided die is rolled 5 times, and each time theresulting sequence of 5 numbers is recorded.(a) How many different sequences are possible? university-logo Jason Aubrey Math 1300 Finite Mathematics
56. 56. Addition Principle Venn Diagrams Multiplication PrincipleExample: A fair 6-sided die is rolled 5 times, and each time theresulting sequence of 5 numbers is recorded.(a) How many different sequences are possible? 6 × 6 × 6 × 6 × 6 = 7776 university-logo Jason Aubrey Math 1300 Finite Mathematics
57. 57. Addition Principle Venn Diagrams Multiplication PrincipleExample: A fair 6-sided die is rolled 5 times, and each time theresulting sequence of 5 numbers is recorded.(a) How many different sequences are possible? 6 × 6 × 6 × 6 × 6 = 7776(b) How many different sequences are possible if all numbersexcept the ﬁrst must be odd? university-logo Jason Aubrey Math 1300 Finite Mathematics
58. 58. Addition Principle Venn Diagrams Multiplication PrincipleExample: A fair 6-sided die is rolled 5 times, and each time theresulting sequence of 5 numbers is recorded.(a) How many different sequences are possible? 6 × 6 × 6 × 6 × 6 = 7776(b) How many different sequences are possible if all numbersexcept the ﬁrst must be odd? 6 × 3 × 3 × 3 × 3 = 486 university-logo Jason Aubrey Math 1300 Finite Mathematics
59. 59. Addition Principle Venn Diagrams Multiplication PrincipleExample: A fair 6-sided die is rolled 5 times, and each time theresulting sequence of 5 numbers is recorded.(a) How many different sequences are possible? 6 × 6 × 6 × 6 × 6 = 7776(b) How many different sequences are possible if all numbersexcept the ﬁrst must be odd? 6 × 3 × 3 × 3 × 3 = 486(c) How many different sequences are possible if the second,third and fourth numbers must be the same? 6 × 6 × 1 × 1 × 6 = 216 university-logo Jason Aubrey Math 1300 Finite Mathematics