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Difficulty: MediumOverlapping Sets and Venn Diagrams

At a technology firm, an evaluation was conducted for 150 software developers to test their proficiency in three programming paradigms: Object-Oriented (OO), Functional (FF), and Reactive (RR). The evaluation revealed that 85 developers are proficient in OO, 60 are proficient in FF, and 45 are proficient in RR. Furthermore, 25 developers are proficient in both OO and FF, 18 are proficient in both FF and RR, 22 are proficient in both OO and RR, and 10 developers are proficient in all three paradigms. How many of the software developers surveyed are proficient in exactly two of these three programming paradigms?

Answer: 35 developers

Answer

The number of software developers proficient in exactly two of the three programming paradigms is 35.
To find the number of developers proficient in exactly two paradigms, subtract the number of developers proficient in all three paradigms (1010) from each of the pairwise totals and sum the results: (25−10)+(18−10)+(22−10)=15+8+12=35(25 - 10) + (18 - 10) + (22 - 10) = 15 + 8 + 12 = 35.

Step-by-Step Solution

1
Identify the total count for each pairwise overlap of paradigms.
The pairwise counts are ∣O∩F∣=25|O \cap F| = 25, ∣F∩R∣=18|F \cap R| = 18, and ∣O∩R∣=22|O \cap R| = 22, with all three paradigms ∣O∩F∩R∣=10|O \cap F \cap R| = 10.
Each given two-set intersection includes individuals who are also proficient in all three paradigms.
2
Subtract the triple intersection from each pairwise total to isolate those proficient in exactly two paradigms.
Proficient in only OO and FF: 25−10=1525 - 10 = 15; proficient in only FF and RR: 18−10=818 - 10 = 8; proficient in only OO and RR: 22−10=1222 - 10 = 12.
The region representing 'exactly two' sets excludes the central region where all three sets overlap.
3
Sum the three distinct exclusive two-set counts.
15+8+12=3515 + 8 + 12 = 35.
Adding these mutually exclusive categories provides the total count of developers in exactly two sets.

Key Concept

Three-Set Venn Diagram Region Calculations

Alternative Method

Using a Venn diagram, enter 1010 in the central triple-intersection region. Next, fill in the three surrounding two-set-only regions by subtracting 1010 from each given pairwise total: 25−10=1525 - 10 = 15, 18−10=818 - 10 = 8, and 22−10=1222 - 10 = 12. Adding these three region counts together gives 15+8+12=3515 + 8 + 12 = 35.
Estimated Time:1m 30s
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Topics

Overlapping Sets and Venn Diagrams

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