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
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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.
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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
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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 = ∅.
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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.
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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.
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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.
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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.
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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?
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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.
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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.
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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
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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
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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)
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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?
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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.
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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 )
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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
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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.
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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.
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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?
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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.
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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:
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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
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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
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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
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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
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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
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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?
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Jason Aubrey Math 1300 Finite Mathematics

30. Addition Principle
Venn Diagrams
Multiplication Principle
Let A and B be sets and A ⊂ U, B ⊂ U,
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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?
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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:
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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?
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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
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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?
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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
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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
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Jason Aubrey Math 1300 Finite Mathematics