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Difficulty: MediumSet Theory Concepts and Venn Diagrams

A media analytics platform surveyed a group of 240240 subscribers regarding the genres of content they watched over the past month: Documentary (DD), Sci-Fi (SS), and Animation (AA). Among the surveyed subscribers:
- 110110 watched Documentary
- 120120 watched Sci-Fi
- 9595 watched Animation
- 4545 watched both Documentary and Sci-Fi
- 4040 watched both Sci-Fi and Animation
- 3535 watched both Documentary and Animation
- 1515 watched none of these three genres

How many of the surveyed subscribers watched content from exactly one of the three genres?

  1. A
    85
  2. B
    105
  3. 145Answer
  4. D
    120
  5. E
    205

Answer

145 subscribers watched content from exactly one genre.
The correct answer is 145. First, subtract the 15 subscribers who watched no genres from the total population of 240 to get 225 subscribers in the union of all three sets. Applying the principle of inclusion-exclusion yields 225=110+120+95−(45+40+35)+∣D∩S∩A∣225 = 110 + 120 + 95 - (45 + 40 + 35) + |D \cap S \cap A|, which solves to ∣D∩S∩A∣=20|D \cap S \cap A| = 20. With 20 subscribers watching all three genres, subscribers watching strictly two genres are 25 (Documentary & Sci-Fi only), 20 (Sci-Fi & Animation only), and 15 (Documentary & Animation only). Isolating subscribers watching only one genre gives 110−(25+15+20)=50110 - (25 + 15 + 20) = 50 for Documentary, 120−(25+20+20)=55120 - (25 + 20 + 20) = 55 for Sci-Fi, and 95−(15+20+20)=4095 - (15 + 20 + 20) = 40 for Animation. Summing these single-genre counts (50+55+4050 + 55 + 40) gives 145.

Step-by-Step Solution

1
Find the number of subscribers who watched at least one genre and apply the 3-set Inclusion-Exclusion Principle to determine the number of subscribers who watched all three genres.
Number watching at least one genre: 240−15=225240 - 15 = 225. By inclusion-exclusion:
∣D∪S∪A∣=∣D∣+∣S∣+∣A∣−(∣D∩S∣+∣S∩A∣+∣D∩A∣)+∣D∩S∩A∣|D \cup S \cup A| = |D| + |S| + |A| - (|D \cap S| + |S \cap A| + |D \cap A|) + |D \cap S \cap A|
225=110+120+95−(45+40+35)+∣D∩S∩A∣225 = 110 + 120 + 95 - (45 + 40 + 35) + |D \cap S \cap A|
225=325−120+∣D∩S∩A∣=205+∣D∩S∩A∣225 = 325 - 120 + |D \cap S \cap A| = 205 + |D \cap S \cap A|
∣D∩S∩A∣=225−205=20|D \cap S \cap A| = 225 - 205 = 20
Knowing the size of the central intersection region (∣D∩S∩A∣=20|D \cap S \cap A| = 20) is necessary to isolate the individual non-overlapping Venn diagram regions.
2
Calculate the number of subscribers who watched EXCLUSIVELY two genres.
Documentary and Sci-Fi only: 45−20=2545 - 20 = 25
Sci-Fi and Animation only: 40−20=2040 - 20 = 20
Documentary and Animation only: 35−20=1535 - 20 = 15
Subtracting the triple intersection from pairwise intersections isolates subscribers who belong to exactly two sets.
3
Calculate the number of subscribers who watched EXACTLY ONE genre and sum them.
Documentary only: 110−(25+15+20)=50110 - (25 + 15 + 20) = 50
Sci-Fi only: 120−(25+20+20)=55120 - (25 + 20 + 20) = 55
Animation only: 95−(15+20+20)=4095 - (15 + 20 + 20) = 40
Total watching exactly one genre: 50+55+40=14550 + 55 + 40 = 145
Subtracting all multi-set overlaps from each genre's total gives the unique single-genre viewers.

Key Concept

Three-Set Inclusion-Exclusion Principle and Regional Segmentation
Estimated Time:1m 45s
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Topics

Set Theory Concepts and Venn Diagrams

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