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Difficulty: MediumSet Theory Concepts and Venn Diagrams

A total of 120 software engineers at a firm were surveyed about their attendance at three professional development workshops: Machine Learning (MM), Cloud Computing (CC), and Cybersecurity (SS). The survey results revealed the following:

- 65 engineers attended Machine Learning
- 55 engineers attended Cloud Computing
- 48 engineers attended Cybersecurity
- 25 engineers attended both Machine Learning and Cloud Computing
- 18 engineers attended both Cloud Computing and Cybersecurity
- 20 engineers attended both Machine Learning and Cybersecurity
- 10 engineers attended all three workshops

How many of the surveyed software engineers attended exactly one of the three workshops?

  1. A
    42
  2. B
    48
  3. 72Answer
  4. D
    105
  5. E
    115

Answer

72 engineers attended exactly one of the three workshops.
To find the number of engineers attending exactly one workshop, calculate the exclusive single-set counts for each category using a 3-set Venn diagram model. Subtracting the overlapping regions gives 30 for Machine Learning only, 22 for Cloud Computing only, and 20 for Cybersecurity only. Summing these exclusive regions yields 30+22+20=7230 + 22 + 20 = 72. Alternatively, using the algebraic formula ∣Exactly 1∣=(∣M∣+∣C∣+∣S∣)−2(∣M∩C∣+∣C∩S∣+∣M∩S∣)+3∣M∩C∩S∣=(65+55+48)−2(25+18+20)+3(10)=168−126+30=72|\text{Exactly 1}| = (|M| + |C| + |S|) - 2(|M \cap C| + |C \cap S| + |M \cap S|) + 3|M \cap C \cap S| = (65 + 55 + 48) - 2(25 + 18 + 20) + 3(10) = 168 - 126 + 30 = 72.

Step-by-Step Solution

1
Find the number of engineers in the regions corresponding to exactly two workshops.
Machine Learning and Cloud Computing only = 25−10=1525 - 10 = 15; Cloud Computing and Cybersecurity only = 18−10=818 - 10 = 8; Machine Learning and Cybersecurity only = 20−10=1020 - 10 = 10.
The given pairwise intersection values include engineers who attended all three workshops.
2
Calculate the number of engineers who attended only one specific workshop.
Machine Learning only = 65−(15+10+10)=3065 - (15 + 10 + 10) = 30; Cloud Computing only = 55−(15+8+10)=2255 - (15 + 8 + 10) = 22; Cybersecurity only = 48−(10+8+10)=2048 - (10 + 8 + 10) = 20.
Subtract all overlapping regions (two-workshop-only and three-workshop) from each set's total.
3
Sum the single-workshop regions to find the total attending exactly one workshop.
30+22+20=7230 + 22 + 20 = 72.
The subsets representing single workshop attendance are disjoint.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Venn Diagram Region Partitioning
Estimated Time:2m 0s
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Topics

Set Theory Concepts and Venn Diagrams

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