Math 1300: Section 7- 3 Basic Counting Principles
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The document discusses basic counting principles for sets, including the addition principle and Venn diagrams. It provides examples of using the addition principle to count the total number of students in a class when given the numbers of students in different categories, like males and females or different majors. It defines the addition principle formula as the total being equal to the sum of the individual set counts minus any overlap between sets.
Math 1300: Section 7- 3 Basic Counting Principles
- 1. Addition Principle Venn Diagrams Multiplication Principle Math 1300 Finite Mathematics Section 7-3: Basic Counting Principles Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
- 2. Addition Principle Venn Diagrams Multiplication Principle Example: If enrollment in a section of Math 1300 consists of 13 males and 15 females, then it is clear that there are a total of 28 students in the class. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 3. Addition Principle Venn Diagrams Multiplication Principle Example: If enrollment in a section of Math 1300 consists of 13 males and 15 females, then it is clear that there are a total of 28 students in the class. To represent this in terms of set operations, we would first assign 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. Addition Principle Venn Diagrams Multiplication Principle Example: If enrollment in a section of Math 1300 consists of 13 males and 15 females, then it is clear that there are a total of 28 students in the class. To represent this in terms of set operations, we would first assign names to the sets. Let M = set of male students in the section F = set of female students in the section Notice that M ∪ F is the set of all students in the class, and that M ∩ F = ∅. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 5. Addition Principle Venn Diagrams Multiplication Principle Example: If enrollment in a section of Math 1300 consists of 13 males and 15 females, then it is clear that there are a total of 28 students in the class. To represent this in terms of set operations, we would first assign names to the sets. Let M = set of male students in the section F = set of female students in the section Notice that M ∪ F is the set of all students in the class, and that M ∩ F = ∅. The total number of students in the class is then represented 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. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose we are told that an economics class consists of 22 business majors and 16 journalism majors. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 7. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose we are told that an economics class consists of 22 business majors and 16 journalism majors. Let B represent the set of business majors and J represent the set of journalism majors. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 8. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose we are told that an economics class consists of 22 business majors and 16 journalism majors. Let B represent the set of business majors and J represent the set of journalism majors. Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of all students in the class. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 9. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose we are told that an economics class consists of 22 business majors and 16 journalism majors. Let B represent the set of business majors and J represent the set of journalism majors. Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of all students in the class. Can we conclude that n(B ∪ J) = 22 + 16 = 38? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 10. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose we are told that an economics class consists of 22 business majors and 16 journalism majors. Let B represent the set of business majors and J represent the set of journalism majors. Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of all students 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. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose we are told that an economics class consists of 22 business majors and 16 journalism majors. Let B represent the set of business majors and J represent the set of journalism majors. Then n(B) = 22 and n(J) = 16, and B ∪ J is the set of all students 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 both business and journalism. If B ∩ J = ∅, then we must avoid counting these students twice. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 12. Addition Principle Venn Diagrams Multiplication Principle If we had, say, 7 double majors in the class, then n(B ∩ J) = 7 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 13. Addition Principle Venn Diagrams Multiplication Principle If we had, say, 7 double majors in the class, then n(B ∩ J) = 7 And 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. Addition Principle Venn Diagrams Multiplication Principle Theorem (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. Addition Principle Venn Diagrams Multiplication Principle Example: A marketing survey of a group of kids indicated that 25 owned a Nintendo DS and 15 owned a Wii. If 10 kids had both a DS and a Wii, how many kids interviewed have a DS or a Wii? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 16. Addition Principle Venn Diagrams Multiplication Principle Example: A marketing survey of a group of kids indicated that 25 owned a Nintendo DS and 15 owned a Wii. If 10 kids had both a DS and a Wii, how many kids interviewed have a DS or a Wii? Let D represent the set of kids with a DS, and let W represent the set of kids with a Wii. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 17. Addition Principle Venn Diagrams Multiplication Principle Example: A marketing survey of a group of kids indicated that 25 owned a Nintendo DS and 15 owned a Wii. If 10 kids had both a DS and a Wii, how many kids interviewed have a DS or a Wii? Let D represent the set of kids with a DS, and let W represent the 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. Addition Principle Venn Diagrams Multiplication Principle Example: A marketing survey of a group of kids indicated that 25 owned a Nintendo DS and 15 owned a Wii. If 10 kids had both a DS and a Wii, how many kids interviewed have a DS or a Wii? Let D represent the set of kids with a DS, and let W represent the 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. Addition Principle Venn Diagrams Multiplication Principle Example: A marketing survey of a group of kids indicated that 25 owned a Nintendo DS and 15 owned a Wii. If 10 kids had both a DS and a Wii, how many kids interviewed have a DS or a Wii? Let D represent the set of kids with a DS, and let W represent the set of kids with a Wii. n(D ∪ W ) = n(D) + n(W ) − n(D ∩ W ) n(D ∪ W ) = 25 + 15 − 10 = 30 Number of kids with a DS or a Wii: 30. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 20. Addition Principle Venn Diagrams Multiplication Principle In problems which involve more than two sets or which involve complements of sets, it is often helpful to draw a Venn Diagram. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 21. Addition Principle Venn Diagrams Multiplication Principle In problems which involve more than two sets or which involve complements of sets, it is often helpful to draw a Venn Diagram. Example: In a certain class, there are 23 majors in Psychology, 16 majors in English and 7 students who are majoring in both Psychology 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. Addition Principle Venn Diagrams Multiplication Principle In problems which involve more than two sets or which involve complements of sets, it is often helpful to draw a Venn Diagram. Example: In a certain class, there are 23 majors in Psychology, 16 majors in English and 7 students who are majoring in both Psychology 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 E represent the set of English majors. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 23. Addition Principle Venn Diagrams Multiplication Principle Then n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are also given 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. Addition Principle Venn Diagrams Multiplication Principle Then n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are also given 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. Addition Principle Venn Diagrams Multiplication Principle Then n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are also given 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. Addition Principle Venn Diagrams Multiplication Principle Then n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are also given 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. Addition Principle Venn Diagrams Multiplication Principle Then n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are also given 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. Addition Principle Venn Diagrams Multiplication Principle Then n(P) = 23, n(E) = 16, and n(P ∩ E) = 7. We are also given 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. Addition Principle Venn Diagrams Multiplication Principle Example: A survey of 100 college faculty who exercise regularly 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. How many of the faculty members do not do any of these three activities? How many just jog? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 30. Addition Principle Venn Diagrams Multiplication Principle Let A and B be sets and A ⊂ U, B ⊂ U, university-logo Jason Aubrey Math 1300 Finite Mathematics
- 31. Addition Principle Venn Diagrams Multiplication Principle Let 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. Addition Principle Venn Diagrams Multiplication Principle Let 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. Addition Principle Venn Diagrams Multiplication Principle Let 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. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following Venn diagram. U A B university-logo Jason Aubrey Math 1300 Finite Mathematics
- 35. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following Venn diagram. U A B 63 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 36. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following Venn diagram. U n(A ∪ B ) = n(A ) + n(B ) − n(A ∩ B ) A B = 81 + 90 − 63 = 108 63 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 37. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following 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 63 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 38. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following 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 ) = 99 63 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 39. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following 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 ) = 99 63 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 40. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following 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 ) = 99 63 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 41. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following 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 ) = 99 63 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 42. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following 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 ) = 99 63 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 43. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose that A and B are sets with n(A ) = 81, n(B ) = 90, n(A ∩ B ) = 63, and n(U) = 180. Determine the number of elements in each of the four disjoint subsets in the following 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 ) = 99 63 n(B) = 180 − n(B ) = 90 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 44. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose you have 4 pairs of pants in your closet, 3 different shirts and 2 pairs of shoes. How many different ways can you choose an outfit consisting of one pair of pants, one shirt and one pair of shoes? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 45. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose you have 4 pairs of pants in your closet, 3 different shirts and 2 pairs of shoes. How many different ways can you choose an outfit consisting of one pair of pants, one shirt and one pair of shoes? Let’s consider this as a sequence of operations: university-logo Jason Aubrey Math 1300 Finite Mathematics
- 46. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose you have 4 pairs of pants in your closet, 3 different shirts and 2 pairs of shoes. How many different ways can you choose an outfit consisting of one pair of pants, one shirt 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. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose you have 4 pairs of pants in your closet, 3 different shirts and 2 pairs of shoes. How many different ways can you choose an outfit consisting of one pair of pants, one shirt 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. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose you have 4 pairs of pants in your closet, 3 different shirts and 2 pairs of shoes. How many different ways can you choose an outfit consisting of one pair of pants, one shirt 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. Addition Principle Venn Diagrams Multiplication Principle Example: Suppose you have 4 pairs of pants in your closet, 3 different shirts and 2 pairs of shoes. How many different ways can you choose an outfit consisting of one pair of pants, one shirt 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 Now for each operation, there is a specified number of ways to perform this operation: Operation Number of Ways O1 N1 = 4 O2 N2 = 3 O3 N3 = 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 50. Addition Principle Venn Diagrams Multiplication Principle So we have i Oi Ni 1 O1 4 2 O2 3 3 O3 2 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 51. Addition Principle Venn Diagrams Multiplication Principle So we have i Oi Ni 1 O1 4 2 O2 3 3 O3 2 Then we can draw a tree diagram to see that there are N1 · N2 · N3 = 4(3)(2) = 24 different outfits. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 52. Addition Principle Venn Diagrams Multiplication Principle Theorem (Multiplication Principle) If two operations O1 and O2 are performed in order, with N1 possible outcomes for the first operation and N2 possible outcomes for the second operation, then there are N1 · N2 possible combined outcomes for the first operation followed by the second. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 53. Addition Principle Venn Diagrams Multiplication Principle Theorem (Generalized Multiplication Principle) In general, if n operations O1 , O2 , · · · , On are performed in order, with possible number of outcomes N1 , N2 , . . . , Nn , respectively, then there are N1 · N2 · · · Nn possible combined outcomes of the operations performed in the given order. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 54. Addition Principle Venn Diagrams Multiplication Principle Example: A fair 6-sided die is rolled 5 times, and each time the resulting sequence of 5 numbers is recorded. university-logo Jason Aubrey Math 1300 Finite Mathematics
- 55. Addition Principle Venn Diagrams Multiplication Principle Example: A fair 6-sided die is rolled 5 times, and each time the resulting sequence of 5 numbers is recorded. (a) How many different sequences are possible? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 56. Addition Principle Venn Diagrams Multiplication Principle Example: A fair 6-sided die is rolled 5 times, and each time the resulting 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. Addition Principle Venn Diagrams Multiplication Principle Example: A fair 6-sided die is rolled 5 times, and each time the resulting 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 numbers except the first must be odd? university-logo Jason Aubrey Math 1300 Finite Mathematics
- 58. Addition Principle Venn Diagrams Multiplication Principle Example: A fair 6-sided die is rolled 5 times, and each time the resulting 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 numbers except the first must be odd? 6 × 3 × 3 × 3 × 3 = 486 university-logo Jason Aubrey Math 1300 Finite Mathematics
- 59. Addition Principle Venn Diagrams Multiplication Principle Example: A fair 6-sided die is rolled 5 times, and each time the resulting 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 numbers except 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