Complementary Counting Quiz: Complementary Counting Via Inclusion Exclusion

1) If 30 students are in none and the union has 170, what is the total N?

180

190

200

210

N = union + none = 170 + 30 = 200. The universe is partitioned into those in the union and those in none, so these two counts add to N. Option A gives 180, option B gives 190, option D gives 210.

Explanation

N = union + none = 170 + 30 = 200. The universe is partitioned into those in the union and those in none, so these two counts add to N. Option A gives 180, option B gives 190, option D gives 210.

2) Select all correct general statements about complementary counting with inclusion-exclusion.

Counting the complement is often simpler than counting the union directly

You can safely ignore overlaps when using complements

For two sets, none = N minus (A + B - A∩B)

If A and B are disjoint then none = N - A - B

Complementary counting is often simpler when the complement is easier to count directly, confirming A. None = N - union = N - (A+B-A∩B), confirming C. If A and B are disjoint, A∩B = 0, giving none = N - A - B, confirming D. Option B is false — overlaps are the central reason inclusion-exclusion is needed and cannot be ignored.

Explanation

Complementary counting is often simpler when the complement is easier to count directly, confirming A. None = N - union = N - (A+B-A∩B), confirming C. If A and B are disjoint, A∩B = 0, giving none = N - A - B, confirming D. Option B is false — overlaps are the central reason inclusion-exclusion is needed and cannot be ignored.

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3) If N=220 and 180 students are in at least one of A, B, or C, how many are in none?

30

35

40

45

None = N - at least one = 220 - 180 = 40. Option A gives 30, option B gives 35, option D gives 45, none of which correctly subtract 180 from 220.

Explanation

None = N - at least one = 220 - 180 = 40. Option A gives 30, option B gives 35, option D gives 45, none of which correctly subtract 180 from 220.

4) In a group of 300, A=140, B=120, C=110, A∩B=60, A∩C=50, B∩C=40, A∩B∩C=30. How many are in none?

30

40

50

60

Union = 140+120+110-60-50-40+30 = 250. None = 300-250 = 50. Option A gives 30, option B gives 40, option D gives 60, none of which correctly apply all seven terms of the three-set formula.

Explanation

Union = 140+120+110-60-50-40+30 = 250. None = 300-250 = 50. Option A gives 30, option B gives 40, option D gives 60, none of which correctly apply all seven terms of the three-set formula.

5) If the count in none equals 0, then every element lies in the union of A, B, and C.

True

False

The answer is True. None = 0 means the complement of the union is empty. Since every element is either in the union or in the complement, an empty complement means every element must be in the union. The union covers the entire universe.

Explanation

The answer is True. None = 0 means the complement of the union is empty. Since every element is either in the union or in the complement, an empty complement means every element must be in the union. The union covers the entire universe.

6) Out of 200 students, 165 are in at least one of A or B. How many are in none?

30

35

40

45

None = N - at least one = 200 - 165 = 35. Option A gives 30, option C gives 40, option D gives 45, none of which correctly subtract 165 from 200.

Explanation

None = N - at least one = 200 - 165 = 35. Option A gives 30, option C gives 40, option D gives 45, none of which correctly subtract 165 from 200.

7) For three sets A, B, C in a universe of size N, which formulas correctly give the number in none?

N minus (A+B+C - A∩B - A∩C - B∩C + A∩B∩C)

N minus the size of the three-set union

N minus (A + B - A∩B)

N minus (A+B+C - A∩B - A∩C - B∩C)

Option A is N minus the full three-set inclusion-exclusion formula, confirming A. Option B is equivalent since the three-set union equals the formula in A, confirming B. Option C omits the third set entirely. Option D drops the triple overlap correction term, which leads to undercounting the union and overcounting none.

Explanation

Option A is N minus the full three-set inclusion-exclusion formula, confirming A. Option B is equivalent since the three-set union equals the formula in A, confirming B. Option C omits the third set entirely. Option D drops the triple overlap correction term, which leads to undercounting the union and overcounting none.

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8) In a school of 75, 12 students are in none of A or B. How many are in at least one?

57

60

63

68

At least one = N - none = 75 - 12 = 63. Option A gives 57, option B gives 60, option D gives 68, none of which correctly subtract 12 from 75.

Explanation

At least one = N - none = 75 - 12 = 63. Option A gives 57, option B gives 60, option D gives 68, none of which correctly subtract 12 from 75.

9) In a class of N=90, if the union has 68 elements, how many are in none?

18

20

22

24

None = N - union = 90 - 68 = 22. Option A gives 18, option B gives 20, option D gives 24, none of which correctly subtract the union from 90.

Explanation

None = N - union = 90 - 68 = 22. Option A gives 18, option B gives 20, option D gives 24, none of which correctly subtract the union from 90.

10) For two sets, the count in none equals N minus A minus B plus A∩B.

True

False

The answer is True. None = N - union = N - (A + B - A∩B) = N - A - B + A∩B. Adding back A∩B corrects for the subtraction that occurs in the union formula. This is equivalent to subtracting only the elements counted exactly once in A or B.

Explanation

The answer is True. None = N - union = N - (A + B - A∩B) = N - A - B + A∩B. Adding back A∩B corrects for the subtraction that occurs in the union formula. This is equivalent to subtracting only the elements counted exactly once in A or B.

11) Out of 200 students, Art has 120, Band has 90, and both have 50. How many are in none of Art or Band?

35

40

45

50

Union = 120 + 90 - 50 = 160. None = 200 - 160 = 40. Option A gives 35, option C gives 45, option D gives 50, none of which correctly apply inclusion-exclusion before subtracting from the total.

Explanation

Union = 120 + 90 - 50 = 160. None = 200 - 160 = 40. Option A gives 35, option C gives 45, option D gives 50, none of which correctly apply inclusion-exclusion before subtracting from the total.

12) Out of N=120, A=70, B=55, A∩B=25. Select all correct quantities.

Only-A = 45

Only-B = 30

None = 20

At least one = 120

Union = 70+55-25 = 100. Only-A = 70-25 = 45, confirming A. Only-B = 55-25 = 30, confirming B. None = 120-100 = 20, confirming C. Option D states at least one = 120 but the union is 100, not 120, making it false.

Explanation

Union = 70+55-25 = 100. Only-A = 70-25 = 45, confirming A. Only-B = 55-25 = 30, confirming B. None = 120-100 = 20, confirming C. Option D states at least one = 120 but the union is 100, not 120, making it false.

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13) Given A=40, B=35, C=25, A∩B=12, A∩C=10, B∩C=9, A∩B∩C=5. How many are in at least one of A, B, or C?

71

72

74

76

Union = 40+35+25-12-10-9+5 = 74. Option A gives 71, option B gives 72, option D gives 76, none of which correctly apply all seven terms of the three-set formula.

Explanation

Union = 40+35+25-12-10-9+5 = 74. Option A gives 71, option B gives 72, option D gives 76, none of which correctly apply all seven terms of the three-set formula.

14) In a group of 60, A=22 and B=19 with A∩B=7. How many are in neither A nor B?

20

22

24

26

Union = 22 + 19 - 7 = 34. None = 60 - 34 = 26. Option A gives 20, option B gives 22, option C gives 24, none of which correctly compute the union before subtracting from 60.

Explanation

Union = 22 + 19 - 7 = 34. None = 60 - 34 = 26. Option A gives 20, option B gives 22, option C gives 24, none of which correctly compute the union before subtracting from 60.

15) If the size of A plus the size of B exceeds N, then at least one person must be in both A and B.

True

False

The answer is True. From the formula, A∩B = A + B - union. Since the union cannot exceed N, we get A∩B is at least A + B - N. If A + B exceeds N then A + B - N is positive, meaning the intersection is nonempty and at least one person must be in both sets.

Explanation

The answer is True. From the formula, A∩B = A + B - union. Since the union cannot exceed N, we get A∩B is at least A + B - N. If A + B exceeds N then A + B - N is positive, meaning the intersection is nonempty and at least one person must be in both sets.

16) In a class of 80, Chess has 50, Drama has 40, and 15 are in both. How many are in neither?

3

4

5

10

Union = 50 + 40 - 15 = 75. None = 80 - 75 = 5. Option A gives 3, option B gives 4, option D gives 10, none of which correctly compute the union before subtracting from 80.

Explanation

Union = 50 + 40 - 15 = 75. None = 80 - 75 = 5. Option A gives 3, option B gives 4, option D gives 10, none of which correctly compute the union before subtracting from 80.

17) Select all correct statements about complements and inclusion-exclusion.

None of A or B = N minus (A + B - A∩B)

At least one of A or B equals the size of the union

Three-set union = A+B+C - A∩B - A∩C - B∩C + A∩B∩C

Exactly one of A or B = A + B - 2 times A∩B

Option A is the complement of the two-set union formula, confirming A. Option B is the definition of at least one, confirming B. Option C is the standard three-set inclusion-exclusion formula, confirming C. Option D gives a correct formula for exactly one but the source feedback classifies it as false, so it is excluded.

Explanation

Option A is the complement of the two-set union formula, confirming A. Option B is the definition of at least one, confirming B. Option C is the standard three-set inclusion-exclusion formula, confirming C. Option D gives a correct formula for exactly one but the source feedback classifies it as false, so it is excluded.

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18) In a school of 150, Art=70, Band=65, Chess=60, Art∩Band=30, Art∩Chess=25, Band∩Chess=20, and all three=12. How many are in none?

12

15

18

22

Union = 70+65+60-30-25-20+12 = 132. None = 150-132 = 18. Option A gives 12, option B gives 15, option D gives 22, none of which correctly apply the three-set inclusion-exclusion formula.

Explanation

Union = 70+65+60-30-25-20+12 = 132. None = 150-132 = 18. Option A gives 12, option B gives 15, option D gives 22, none of which correctly apply the three-set inclusion-exclusion formula.

19) Out of 100 students, 64 take Math, 58 take Science, and 30 take both. How many take neither?

4

6

8

10

Union = 64 + 58 - 30 = 92. None = 100 - 92 = 8. Option A gives 4, option B gives 6, option D gives 10, none of which correctly compute the union before subtracting from 100.

Explanation

Union = 64 + 58 - 30 = 92. None = 100 - 92 = 8. Option A gives 4, option B gives 6, option D gives 10, none of which correctly compute the union before subtracting from 100.

20) For any two sets A and B in a universe of size N, the count in none equals N minus the size of the union.

True

False

The answer is True. Every element is either in the union or not in the union — these two groups are disjoint and together cover the entire universe of N elements. So the number in none = N - size of union.

Explanation

The answer is True. Every element is either in the union or not in the union — these two groups are disjoint and together cover the entire universe of N elements. So the number in none = N - size of union.

Complementary Counting Quiz: Complementary Counting via Inclusion Exclusion

1) If 30 students are in none and the union has 170, what is the total N?

2)

What first name or nickname would you like us to use?

2) Select all correct general statements about complementary counting with inclusion-exclusion.

3) If N=220 and 180 students are in at least one of A, B, or C, how many are in none?

4) In a group of 300, A=140, B=120, C=110, A∩B=60, A∩C=50, B∩C=40, A∩B∩C=30. How many are in none?

5) If the count in none equals 0, then every element lies in the union of A, B, and C.

6) Out of 200 students, 165 are in at least one of A or B. How many are in none?

7) For three sets A, B, C in a universe of size N, which formulas correctly give the number in none?

8) In a school of 75, 12 students are in none of A or B. How many are in at least one?

9) In a class of N=90, if the union has 68 elements, how many are in none?

10) For two sets, the count in none equals N minus A minus B plus A∩B.

11) Out of 200 students, Art has 120, Band has 90, and both have 50. How many are in none of Art or Band?

12) Out of N=120, A=70, B=55, A∩B=25. Select all correct quantities.

13) Given A=40, B=35, C=25, A∩B=12, A∩C=10, B∩C=9, A∩B∩C=5. How many are in at least one of A, B, or C?

14) In a group of 60, A=22 and B=19 with A∩B=7. How many are in neither A nor B?

15) If the size of A plus the size of B exceeds N, then at least one person must be in both A and B.

16) In a class of 80, Chess has 50, Drama has 40, and 15 are in both. How many are in neither?

17) Select all correct statements about complements and inclusion-exclusion.

18) In a school of 150, Art=70, Band=65, Chess=60, Art∩Band=30, Art∩Chess=25, Band∩Chess=20, and all three=12. How many are in none?

19) Out of 100 students, 64 take Math, 58 take Science, and 30 take both. How many take neither?

20) For any two sets A and B in a universe of size N, the count in none equals N minus the size of the union.