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Difficulty: MediumLogical Venn Diagrams

In a state policy research institute, a survey of 200200 research analysts was conducted regarding their domain specializations in three sectors: Health (HH), Education (EE), and Agriculture (AA). The data revealed that 9090 analysts specialize in Health, 8585 in Education, and 8080 in Agriculture. Furthermore, 3535 analysts specialize in both Health and Education, 3030 in both Education and Agriculture, 2525 in both Health and Agriculture, and 1010 analysts specialize in all three sectors. How many research analysts specialize in corporate policy research analysts who specialize in exactly two of these sectors?

  1. 6060Answer
  2. B
    7070
  3. C
    8080
  4. D
    9090

Answer

The number of research analysts who specialize in exactly two sectors is 6060.
To find the number of analysts specializing in exactly two sectors, we must determine the exclusive intersection of each pair of sectors. The given intersections represent analysts in at least two sectors.
- Analysts specializing in ONLY Health and Education = 35−10=2535 - 10 = 25
- Analysts specializing in ONLY Education and Agriculture = 30−10=2030 - 10 = 20
- Analysts specializing in ONLY Health and Agriculture = 25−10=1525 - 10 = 15

Summing these exclusive regions gives 25+20+15=6025 + 20 + 15 = 60.

Step-by-Step Solution

1
Identify given set values and intersections
n(H∩E)=35n(H \cap E) = 35, n(E∩A)=30n(E \cap A) = 30, n(H∩A)=25n(H \cap A) = 25, and n(H∩E∩A)=10n(H \cap E \cap A) = 10.
Each pairwise intersection given includes both analysts specializing in exactly two sectors and analysts specializing in all three sectors.
2
Calculate the number of analysts specializing ONLY in each specific pair of two sectors
Only Health & Education = 35−10=2535 - 10 = 25; Only Education & Agriculture = 30−10=2030 - 10 = 20; Only Health & Agriculture = 25−10=1525 - 10 = 15.
Subtracting n(H∩E∩A)n(H \cap E \cap A) isolates the analysts specializing in exactly two sectors for each pair.
3
Sum the three isolated regions of exactly two specializations
Total = 25+20+15=6025 + 20 + 15 = 60.
Adding these three mutually exclusive regions yields the overall count of analysts specializing in exactly two sectors.

Key Concept

Logical Venn Diagrams - Intersection Exclusion Principle

Alternative Method

Using the general formula: Exactly two sets=n(H∩E)+n(E∩A)+n(H∩A)−3×n(H∩E∩A)=35+30+25−3(10)=90−30=60\text{Exactly two sets} = n(H \cap E) + n(E \cap A) + n(H \cap A) - 3 \times n(H \cap E \cap A) = 35 + 30 + 25 - 3(10) = 90 - 30 = 60.
Estimated Time:1m 30s
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Topics

Logical Venn Diagrams

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