1. Section 16 Inclusion/Exclusion
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MATH 106, Section 16

2. Set Notation A set is any well-defined collection of objects.
U represents the universal set (i.e., the universe).
A  B means that
A  B represents
A  B represents
~A represents
#A represents
A is a subset of B, that is, every item in A must also be an item in B. (Every set is a subset of U.)
the intersection of A and B, that is, the set of all items that are in both A and B.
the union of A and B, that is, the set of all items that are in at least one of A or B.
the complement of A, that is, the set of all items that are not in A.
the size of A, that is, the number of items that are in A.
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MATH 106, Section 16

3. #1
Consider a universal set consisting of the positive integers up to (and including) 12, that is, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. We define the subsets
A = multiples of 2
B = multiples of 3
C = multiples of 4
D = multiples of 5
Use set notation to display each subset; then find each of the following:
#A = #B =
#C = #D =
~A = ~B =
~C = ~D =
{2, 4, 6, 8, 10, 12}
{3, 6, 9, 12}
{4, 8, 12}
{5, 10}
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MATH 106, Section 16

4. Consider a universal set consisting of the positive integers up to (and including) 12, that is, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. We define the subsets
A = multiples of 2
B = multiples of 3
C = multiples of 4
D = multiples of 5
Use set notation to display each subset; then find each of the following:
#A = #B =
#C = #D =
~A = ~B =
~C = ~D =
{2, 4, 6, 8, 10, 12}
{3, 6, 9, 12}
{4, 8, 12}
{5, 10} 6 4 3 2
{1, 3, 5, 7, 9, 11}
{1, 2, 4, 5, 7. 8, 10, 11}
{1, 2, 3, 5, 6, 7, 9, 10, 11}
{1, 2, 3, 4, 6, 7, 8, 9, 11, 12}
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MATH 106, Section 16

5. AB = AC = BD = AB = AC = BD =
In each section of the Venn diagrams displayed, list the members of the set that section represents.
{6, 12}
{4, 8, 12}
{ } or 
{2, 3, 4, 6, 8, 9, 10, 12}
{2, 4, 6, 8, 10, 12}
{3, 5, 6, 9, 10, 12} 1 5 7 11 2 4 8 10 6 12 3 9 A B
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MATH 106, Section 16

6. 1 3 5 7 9 11 2 6 10 4 8 12 A C 1 2 4 7 8 11 3 6 9 12 5 10 B D
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MATH 106, Section 16

7. Complete each of the following formulas, and then verify that each is true:
#(AB) = #A + #B
#(AC) = #A + #C
#(BD) = #B + #D
– #(AB) 8 6 4 2
– #(AC) 6 6 3 3
When we continue with this handout, we shall consider this same type of situation with three sets A, B, and C. Right now, we return to the Subsets, Strings, Equations Handout.
– #(BD) 6 4 2
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MATH 106, Section 16

8. Use the rest of this class period to work on #1 on the Subsets, Strings, Equations Handout, which must be submitted for homework either at the end of this class or in the class indicated on the course schedule.
As you work on each problem, check to see if your final answer is correct. We will go over in class any problems there are questions about. There will also be some time this period for questions.
CHECK YOUR ANSWERS:
2. (a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l) 84 35 4 7
262,144
6765
6765 2
255,379
262,142
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MATH 106, Section 10

9. NAME__________________________________________________2.
(a)
(b)
A child is playing with large box of blocks that are identical, except that some are red and some are blue.
How many different ways can six red blocks and three blue blocks be arranged in a row? 9!
——– = 84
6! 3!
How many different ways can six red blocks and three blue blocks be arranged in a row so that none of the red blocks are adjacent?
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MATH 106, Section 10

10. (c)
(d)
How many different ways can six red blocks and three blue blocks be arranged in a row so that none of the blue blocks are adjacent? 7!
——– = 35
3! 4!
How many different ways can six red blocks and three blue blocks be arranged in a row so that all of the red blocks are adjacent? 4
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MATH 106, Section 10

11. 2.-continued
(e)
(f)
How many different ways can six red blocks and three blue blocks be arranged in a row so that all of the blue blocks are adjacent? 7
How many different ways can six red blocks and three blue blocks be arranged in a row so that blocks of the same color are not adjacent?
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MATH 106, Section 10

12. (g)
(h)
How many different ways can 18 blocks, each one either red or blue, be arranged in a row?
218 = 262,144
How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that none of the blue blocks are adjacent?
F(18) = 6765
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MATH 106, Section 10

13. 2.-continued
(i)
(j)
How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that none of the red blocks are adjacent?
F(18) = 6765
How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that blocks of the same color are not adjacent? 2
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MATH 106, Section 10

14. (k)
(l)
How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that at least two of the red blocks are adjacent?
218 – F(18) = 262,144 – 6765 = 255,379
How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that at least two blocks of the same color are adjacent?
218 – 2 = 262,144 – 2 = 262,142
In the class after the exam, we shall begin with #2 on the Section #16 Handout.
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MATH 106, Section 10

15. Problem #3: Use #U – #(AB) = #U – [#A + #B – #(AB)]
In the class after the exam, we shall begin with #2 on the Section #16 Handout. For next class, do the following problems in the Section 16 Homework:
Problem #3: Use #U – #(AB) = #U – [#A + #B – #(AB)]
freshmen taking mathematics
all freshmen
freshmen taking computer science
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MATH 106, Section 10

16. #2
Consider a universal set consisting of the positive integers up to (and including) 30, that is, U = {1, 2, …, 29, 30}. We define the subsets
A = multiples of 2
B = multiples of 3
C = multiples of 5
Use set notation to display each subset. In each section of the Venn diagram displayed, list the members of the set that section represents. Then find each of the following:
{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}
{3, 6, 9, 12, 15, 18, 21, 24, 27, 30}
{5, 10, 15, 20, 25, 30}
#A = #B =
#C = #(AB) =
#(AC) = #(BC) =
#(ABC) = 15 10 6
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MATH 106, Section 16

17. 1 7 11 13 17 19 23 29 2 4 8 14 6 12 18 24 3 9 21 27 16 22 26 28 30 A 10 20 15 B 5 25 C
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MATH 106, Section 16

18. Complete the following formula, and then verify that it is true:
#(ABC) =
#A + #B + #C
– #(AB)
– #(AC)
– #(BC)
+ #(ABC) 15 10 6 5 3 2 1
This is 22, which is what we count for ABC from the Venn diagram.
#A = #B =
#C = #(AB) =
#(AC) = #(BC) =
#(ABC) = 15 10 6 5 3 2 1
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MATH 106, Section 16

19. U = {1, 2, …, 29, 30}
A = multiples of 2 B = multiples of 3 C = multiples of 5
Note the transition from words to set notation and vice versa:
= multiples of each (all) of 2 and 3 and 5
= multiples of (at least) one of 2 or 3 or 5
= multiples of none of 2 and 3 and 5
= not a multiple of (at least) one of 2 or 3 or 5
ABC
ABC
~(ABC)
OR (~A)(~B)(~C)
~(ABC)
OR (~A)(~B)(~C)

20. A = the set of arrangements where the letters E, F are together #3
Let the set of all different possible arrangements (permutations) of the letters EFNOPZ be the universal set. We define the subsets
A = the set of arrangements where the letters E, F are together
B = the set of arrangements where the letters N, O, P are together
(a)
(b)
Find the size of the universe.
#U =
6! = 720
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the letters E, F are
not together
the set of arrangements where the letters N, O, P are
not together
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MATH 106, Section 16

21. A = the set of arrangements where the letters E, F are together
Let the set of all different possible arrangements (permutations) of the letters EFNOPZ be the universal set. We define the subsets
A = the set of arrangements where the letters E, F are together
B = the set of arrangements where the letters N, O, P are together
(a)
(b)
Find the size of the universe.
#U =
6! = 720
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the letters E, F are
not together
the set of arrangements where the letters N, O, P are
not together
the set of arrangements where the letters E, F are together
the letters N, O, P are together
and
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MATH 106, Section 16

22. the set of arrangements where the letters E, F are together
~(AB)
AB
~(AB)
the set of arrangements where the letters E, F are not together the letters N, O, P are not together or
the set of arrangements where the letters E, F are together
the letters N, O, P are together or
the set of arrangements where the letters E, F are not together the letters N, O, P are not together
and
(c)
(d)
Find each of the following:
#A = #B =
5!  2! =
240
4!  3! =
144
Find the number of arrangements where the letters E, F are together and the letters N, O, P are together.
#(AB) =
3!  2!  3! = 72
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MATH 106, Section 16

23. (e)
(f)
Find the number of arrangements where the letters E, F are together or the letters N, O, P are together.
#(AB) =
#A + #B – #(AB) =
– 72 =
312
Find the number of arrangements where the letters E, F are not together and the letters N, O, P are not together.
#~(AB) =
#U – #(AB) =
720 – 312 =
408
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MATH 106, Section 16

24. B = the set of all arrangements where the letters O,P,Q,R are together #4
Let the set of all different possible arrangements (permutations) of the letters EFGHIOPQRUVW be the universal set. We define the subsets
A = the set of all arrangements where the letters E,F,G,H,I are together
B = the set of all arrangements where the letters O,P,Q,R are together
C = the set of all arrangements where the letters U,V,W are together
(a)
(b)
Find the size of the universe.
#U =
12! = 479,001,600
Describe in words each of the sets listed. ~A ~B ~C
set of arrangements where letters E, F, G, H, I are
not together
set of arrangements where letters O, P, Q, R are
not together
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MATH 106, Section 16

25. B = the set of all arrangements where the letters O,P,Q,R are together
Let the set of all different possible arrangements (permutations) of the letters EFGHIOPQRUVW be the universal set. We define the subsets
A = the set of all arrangements where the letters E,F,G,H,I are together
B = the set of all arrangements where the letters O,P,Q,R are together
C = the set of all arrangements where the letters U,V,W are together
(a)
(b)
Find the size of the universe.
#U =
12! = 479,001,600
Describe in words each of the sets listed. ~A ~B ~C
set of arrangements where letters E, F, G, H, I are
not together
set of arrangements where letters O, P, Q, R are
not together
set of arrangements where letters U, V, W are
not together
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MATH 106, Section 16

26. AB
AC
BC
ABC
ABC
the set of arrangements where the letters E, F, G, H, I are together the letters O, P, Q, R are together
and
the set of arrangements where the letters E, F, G, H, I are together the letters U, V, W are together
and
the set of arrangements where the letters O, P, Q, R are together the letters U, V, W are together
and
the set of arrangements where the letters E, F, G, H, I are together the letters O, P, Q, R are together the letters U, V, W are together
and
and
the set of arrangements where the letters E, F, G, H, I are together the letters O, P, Q, R are together the letters U, V, W are together or or
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MATH 106, Section 16

27. Find each of the following: #A = #B =
#C = #(AB) =
#(AC) = #(BC) =
#(ABC) =
8!  5! = 4,838,400
9!  4! = 8,709,120
10!  3! = 21,772,800
5!  5!  4! = 345,600
6!  5!  3! = 518,400
7!  4!  3! = 725,760
3!  5!  4!  3! = 103,680
Find the number of arrangements where the letters E, F, G, H, I are together or the letters O, P, Q, R are together or the letters U, V, W are together.
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MATH 106, Section 16

28. (d)
Find the number of arrangements where the letters E, F, G, H, I are together or the letters O, P, Q, R are together or the letters U, V, W are together.
#(ABC) =
#A + #B + #C
– #(AB)
– #(AC)
– #(BC)
+ #(ABC)
= 4,838, ,709, ,772,800
– 345,600 – 518,400 – 725, ,680 =
33,834,240
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MATH 106, Section 16

29. Problem #4: Use #(ABC) =
For next class, do Problem #5 on the Section 16 Handout, and do the following problems in the Section 16 Homework:
Problem #4: Use #(ABC) =
#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)
children who play tennis
children who play soccer
children who play baseball
Problem #5: Use #U – #(AB) = #U – [#A + #B – #(AB)]
all 5-card hands from 52 cards
all 5-card hands from the 48 non-king cards
all 5-card hands from the 48 non-ace cards
all 5-card hands from the 44 non-ace and non-king cards
As time permits, let’s begin work on Problem #5 on the Section 16 Handout. 
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MATH 106, Section 16

30. A = the set of arrangements where the digits 0, 1 are together #5
Let the set of all different possible arrangements (permutations) of the digits be the universal set. We define the subsets
A = the set of arrangements where the digits 0, 1 are together
B = the set of arrangements where the digits 5, 6 are together
(a)
(b)
Find the size of the universe.
#U =
10! = 3,628,800
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the digits 0, 1 are not together
the set of arrangements where the digits 5, 6 are not together
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MATH 106, Section 16

31. A = the set of arrangements where the digits 0, 1 are together
Let the set of all different possible arrangements (permutations) of the digits be the universal set. We define the subsets
A = the set of arrangements where the digits 0, 1 are together
B = the set of arrangements where the digits 5, 6 are together
(a)
(b)
Find the size of the universe.
#U =
10! = 3,628,800
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the digits 0, 1 are not together
the set of arrangements where the digits 5, 6 are not together
the set of arrangements where the digits 0, 1 are together and the digits 5, 6 are together
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MATH 106, Section 16

32. Find each of the following: #A = #B = 9!  2! = 725,760 9!  2! =
the set of arrangements where the digits 0, 1 are not together or the digits 5, 6 are not together
the set of arrangements where the digits 0, 1 are together or the digits 5, 6 are together
the set of arrangements where the digits 0, 1 are not together and the digits 5, 6 are not together
(c)
(d)
Find each of the following:
#A = #B =
9!  2! =
725,760
9!  2! =
725,760
Find the number of arrangements where the digits 0, 1 are together and the digits 5, 6 are together.
#(AB) =
8!  2!  2! =
161,280
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MATH 106, Section 16

33. (e)
(f)
Find the number of arrangements where the digits 0, 1 are together or the digits 5, 6 are together.
#(AB) =
#A + #B – #(AB) =
725, ,760 – 161,280 =
1,290,240
Find the number of arrangements where the digits 0, 1 are not together and the digits 5, 6 are not together.
#~(AB) =
#U – #(AB) =
3,628,800 – 1,290,240 =
2,338,560
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MATH 106, Section 16

34. A = the set of integers where the digit 5 does not appear #6
Let the set of all integers from to be the universal set, where we use leading zeros as needed, i.e., we write integers such as 538 as We define the subsets
A = the set of integers where the digit 5 does not appear
B = the set of integers where the digit 6 does not appear
(a)
(b)
Find the size of the universe.
#U =
106 = 1,000,000
Describe in words each of the sets listed. ~A ~B
the set of integers where the digit 5
appears at least once
the set of integers where the digit 6
appears at least once
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MATH 106, Section 16

35. the set of integers where the digit 5 does not appear
AB
~(AB)
AB
~(AB)
the set of integers where the digit 5 does not appear
the digit 6 does not appear
and
neither of the digits 5 and 6 appear
the set of integers where the digit 5 appears at least once
the digit 6 appears at least once or
at least one of the digits 5 and 6 appears at least once
the set of integers where the digit 5 does not appear
the digit 6 does not appear or
at least one of the digits 5 and 6 does not appear
the set of integers where the digit 5 appears at least once
the digit 6 appears at least once
and
each of the digits 5 and 6 appears at least once
(c)
Find each of the following:
#A = #B =
96 = 531,441
96 = 531,441
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MATH 106, Section 16

36. #U – #(A  B) = #U – [#A + #B – #(A  B)] =
(d)
(e)
Find the number of integers where neither of the digits 5 and 6 appear.
#(AB) =
86 = 262,144
Find the number of integers where the digits 5 and 6 each appears at least once.
#~(A  B) =
#U – #(A  B) = #U – [#A + #B – #(A  B)] =
106 – [ – 86] = 199,262
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MATH 106, Section 16

37. A = the set of integers divisible by 10 #7
Let the set of all integers from 1 to 1000 be the universal set. We define the subsets
A = the set of integers divisible by 10
B = the set of integers divisible by 15
C = the set of integers divisible by 25
(a)
(b)
{10, 20, 30, …, 990, 1000}
{15, 30, 45, …, 975, 990}
{25, 50, 75, …, 975, 1000}
Find the size of the universe.
#U =
1000
Describe in words each of the sets listed. ~A ~B ~C
set of integers not divisible by 10
set of integers not divisible by 15
set of integers not divisible by 25
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MATH 106, Section 16

38. the set of integers divisible by both 10 and 15 10 = 25 15 = 35
AB
AC
BC
ABC
ABC
the set of integers divisible by
both 10 and 15
10 = 25
15 = 35
Integers divisible by each of 10 and 15 must be divisible by
235 = 30
the set of integers divisible by
both 10 and 25
10 = 25
25 = 55
Integers divisible by each of 10 and 25 must be divisible by
255 = 50
the set of integers divisible by
both 15 and 25
15 = 35
25 = 55
Integers divisible by each of 15 and 25 must be divisible by
355 = 75
the set of integers divisible by
each of 10, 15, and 25
10 = 25
15 = 35
25 = 55
Integers divisible by all of 10, 15, and 25 must be divisible by
2355 =
150
the set of integers divisible by
at least one of 10, 15, and 25
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MATH 106, Section 16

39. Find each of the following: #A =
#B =
#C =
#(AB) =
100
110 = 10, 210 = 20, 310 = 30, …,
10010 = 1000 66
115 = 15, 215 = 30, 315 = 45, …,
6615 = 990, 6715 = 1005 40
125 = 25, 225 = 50, 325 = 75, …,
4025 = 1000 33
10 = 25
15 = 35
Integers divisible by each of 10 and 15 must be divisible by
235 = 30
130 = 30, 230 = 60, 330 = 90, …,
3330 = 990, 3430 = 1020
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MATH 106, Section 16

40. Integers divisible by each of 10 and 25 must be divisible by 255 =
#(BC) =
#(ABC) = 20
10 = 25
25 = 55
Integers divisible by each of 10 and 25 must be divisible by
255 = 50
150 = 50, 250 = 100, 350 = 150, …,
2050 = 1000 13
15 = 35
25 = 55
Integers divisible by each of 15 and 25 must be divisible by
355 = 75
175 = 75, 275 = 150, 375 = 225, …,
1375 = 975, 1475 = 1050 6
10 = 25
15 = 35
25 = 55
Integers divisible by all of 10, 15, and 25 must be divisible by
2355 =
150
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MATH 106, Section 16

41. (d)
(e)
Find the number of integers which are divisible by all of 10, 15, or 25.
#(ABC) = 6
Find the number of integers which are divisible by at least one of 10, 15, or 25.
#(ABC) =
#A + #B + #C
– #(AB)
– #(AC)
– #(BC)
+ #(ABC) =
– 33 – 20 – 13 +
6 =
146
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MATH 106, Section 16

42. (f)
Find the number of integers which are divisible by none of 10, 15, or 25.
#~(ABC) =
#U – #(ABC) =
1000 – 146 = 854
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MATH 106, Section 16

43. For next class, do the following problems in the Section 16 Homework:
integers divisible by 15
integers divisible by 12
integers divisible by 18
Problem #6: Use #U – #(ABC) =
#U – [#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)]
You will need to know that 12 = 223,
15 = 35,
18 = 233.
integers with no digit 4
integers with no digit 3
integers with no digit 5
Problem #7: Use #U – #(ABC) =
#U – [#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)]
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MATH 106, Section 16