Discrete Math Quiz: Inclusion Exclusion Principle

7 questions · exam conditions

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Inclusion Exclusion PrincipleQuestion 1 of 7

A survey of 200 students found that 85 take Math, 92 take Science, 78 take English, 35 take both Math and Science, 42 take both Math and English, 38 take both Science and English, and 18 take all three subjects. If the survey counted every student exactly once, how many students take none of these three subjects?

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Discrete Math Quiz

Discrete Math Quiz: Inclusion Exclusion Principle

Practice Inclusion Exclusion Principle in Discrete Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Inclusion Exclusion Principle, giving you a quick way to practice the rules, question types, and explanations that matter most for Discrete Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

32 (correct answer)
25
18
43

Explanation: Using inclusion-exclusion: |M ∪ S ∪ E| = 85 + 92 + 78 - 35 - 42 - 38 + 18 = 158. Students taking none = 200 - 158 = 32. Choice B results from forgetting to add back the triple intersection. Choice C incorrectly uses just the triple intersection. Choice D results from computational errors in the inclusion-exclusion formula.

Question 2

A data scientist analyzes three machine learning model characteristics: High Accuracy (H), Fast Training (F), and Low Memory Usage (L). In a dataset of 320 models, she finds that models with at least one of these characteristics number 275. She also determines that 85 models have exactly one characteristic, 125 have exactly two characteristics, and the remaining models with these characteristics have all three. If she randomly selects a model from those having at least two characteristics, what is the probability it has all three characteristics?

$\frac{125}{190}$
$\frac{65}{125}$
$\frac{65}{190}$ (correct answer)
$\frac{45}{170}$

Explanation: When you encounter probability questions involving sets with overlapping characteristics, you need to carefully identify the sample space and favorable outcomes using set theory principles. Let's organize the given information: 275 models have at least one characteristic, with 85 having exactly one, 125 having exactly two, and the remainder having all three. Since 85 + 125 = 210, the models with all three characteristics number 275 - 210 = 65. The question asks for the probability that a randomly selected model from those having "at least two characteristics" has all three. Models with at least two characteristics include those with exactly two (125 models) plus those with all three (65 models), giving us 125 + 65 = 190 total models in our sample space. The probability is therefore

\frac{\text{models with all three}}{\text{models with at least two}} = \frac{65}{190}

, which is answer C. Looking at the wrong answers: A)

\frac{125}{190}

incorrectly uses 125 (models with exactly two) in the numerator instead of 65. B)

\frac{65}{125}

uses the correct numerator but wrong denominator, considering only models with exactly two characteristics rather than at least two. D)

\frac{45}{170}

appears to involve calculation errors in determining both the number of models with all three characteristics and the total with at least two. Remember: in conditional probability problems, always clearly define your sample space first. Here, "at least two" means exactly two OR all three, not just exactly two.

Question 3

A coding bootcamp tracks student completion of three certification tracks: Web Development (W), Data Science (D), and Mobile Development (M). Among 150 students, 12 completed all three tracks, 25 completed both W and D but not M, 18 completed both W and M but not D, 15 completed both D and M but not W, 30 completed only W, 20 completed only D, and 22 completed only M. How many students completed at least two tracks?

46
58
82
70 (correct answer)

Explanation: When you encounter problems about overlapping groups, you're dealing with set theory and inclusion-exclusion principles. The key is organizing the given information systematically and understanding what "at least two tracks" means. Let's identify each group from the problem: 12 students completed all three tracks (W∩D∩M), 25 completed both W and D but not M, 18 completed both W and M but not D, and 15 completed both D and M but not W. Students who completed "at least two tracks" include all of these categories. To find the answer, simply add up all students in these overlapping categories:

12 + 25 + 18 + 15 = 70

students completed at least two tracks. Choice A (46) represents a common error where students might forget to include one of the two-track categories, perhaps omitting the 12 students who completed all three tracks and the 12 who completed D and M but not W. Choice B (58) likely results from excluding the "all three tracks" group entirely, counting only

25 + 18 + 15 = 58

. Choice C (82) appears to incorrectly include some students who completed only one track, violating the "at least two" requirement. The correct answer is D (70). When tackling set problems, always start by clearly defining what you're looking for. "At least two" means two or more, so include all intersection regions with two or three sets. Draw a Venn diagram if helpful, and double-check that you're not double-counting or missing any relevant categories.

Question 4

A streaming service analyzes viewing patterns for three genres: Action (A), Comedy (C), and Drama (D). Among 1000 subscribers, they find that 680 watch at least one of these genres. The number watching exactly two genres is 180, and 95 watch all three genres. If 285 watch Action, 245 watch Comedy, and 310 watch Drama, how many watch exactly one genre?

385
320
275
405 (correct answer)

Explanation: When you encounter problems about overlapping categories like this, you're dealing with the principle of inclusion-exclusion, which helps organize complex counting scenarios involving sets. To find those watching exactly one genre, work systematically through the given information. You know 680 watch at least one genre, 180 watch exactly two genres, and 95 watch all three. Since the total watching at least one genre includes those watching exactly one, exactly two, and all three genres, you can write:

680 = \text{exactly one} + 180 + 95

Solving: exactly one =

680 - 180 - 95 = 405

You can verify this using inclusion-exclusion. The sum of individual genre watchers (285 + 245 + 310 = 840) counts overlaps multiple times. Those watching exactly two genres are counted twice in this sum, and those watching all three are counted three times. So:

840 = 405 + 2(180) + 3(95) = 405 + 360 + 285 = 1050

, which reconciles with our calculation when accounting for the overlaps. Answer choice A (385) results from incorrectly subtracting only the 95 all-three watchers. Choice B (320) comes from miscalculating the total overlap. Choice C (275) likely stems from confusing the exactly-two count with another calculation step. The correct answer is D (405). Strategy tip: In inclusion-exclusion problems, always clearly distinguish between "exactly" and "at least" language. Set up your equation methodically: total = exactly one + exactly two + exactly three, then solve for the unknown.

Question 5

In a computer network, let $A$ be the set of nodes that can reach server X, $B$ be the set of nodes that can reach server Y, and $C$ be the set of nodes that can reach server Z. If $|A| = 45$ , $|B| = 38$ , $|C| = 52$ , $|A \cap B| = 15$ , $|A \cap C| = 22$ , $|B \cap C| = 18$ , and $|A \cap B \cap C| = 8$ , what is $|A \cup B \cup C|$ ?

112 (correct answer)
135
120
127

Explanation: By inclusion-exclusion: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| = 45 + 38 + 52 - 15 - 22 - 18 + 8 = 112. Choice B adds all individual sets and intersections without proper subtraction. Choice C forgets to add back the triple intersection. Choice D makes an arithmetic error in the calculation.

Question 6

In a database of 500 records, let $A$ be records containing keyword "algorithm", $B$ be records containing "optimization", and $C$ be records containing "machine learning". If $|A^c \cap B^c \cap C^c| = 125$ and the inclusion-exclusion calculation gives $|A \cup B \cup C| = 375$ , which statement must be true?

The inclusion-exclusion calculation contains an error since 375 + 125 ≠ 500
Exactly 125 records contain none of the three keywords, confirming the calculation is consistent (correct answer)
The sets A, B, and C must be pairwise disjoint for this result to hold
There must be exactly 250 records in the intersection $A \cap B \cap C$

Explanation: Since the total is 500 records, and |A ∪ B ∪ C| = 375, then |A^c ∩ B^c ∩ C^c| = 500 - 375 = 125, which matches the given value. This confirms the calculation is consistent. Choice A incorrectly suggests an error when 375 + 125 = 500 exactly. Choice C is false since overlapping sets can produce this result. Choice D makes an unfounded claim about the triple intersection size.

Question 7

In a cybersecurity audit, three types of vulnerabilities are checked: Authentication (A), Encryption (E), and Access Control (C). A system can have multiple vulnerability types. If $P(A) = 0.3$ , $P(E) = 0.25$ , $P(C) = 0.4$ , $P(A \cap E) = 0.08$ , $P(A \cap C) = 0.15$ , $P(E \cap C) = 0.12$ , and $P(A \cap E \cap C) = 0.05$ , what is the probability that a randomly selected system has none of these vulnerabilities?

0.40
0.45
0.35 (correct answer)
0.50

Explanation: When you encounter a problem asking for the probability that none of several events occur, you're dealing with the complement of the union of those events. The key insight is to use the formula:

P(\text{none}) = 1 - P(A \cup E \cup C)

. To find

P(A \cup E \cup C)

, you need the inclusion-exclusion principle:

P(A \cup E \cup C) = P(A) + P(E) + P(C) - P(A \cap E) - P(A \cap C) - P(E \cap C) + P(A \cap E \cap C)

. Substituting the given values:

P(A \cup E \cup C) = 0.3 + 0.25 + 0.4 - 0.08 - 0.15 - 0.12 + 0.05 = 0.95 - 0.35 + 0.05 = 0.65

. Therefore,

P(\text{none}) = 1 - 0.65 = 0.35

, which is answer C. Answer A (0.40) likely comes from incorrectly calculating

1 - P(C) = 1 - 0.4

, mistakenly focusing only on the largest individual probability. Answer B (0.45) might result from computational errors in the inclusion-exclusion formula, such as forgetting to add back the triple intersection. Answer D (0.50) could come from oversimplifying the problem or making multiple arithmetic mistakes. Remember that "none of the events" problems always involve complements and unions. Write out the inclusion-exclusion formula carefully, paying special attention to the alternating signs, and double-check your arithmetic. The inclusion-exclusion principle is fundamental in probability and appears frequently on discrete math exams.