In a state administrative audit of 300 urban development projects, participation across th… — State PSC Exam

In a state administrative audit of $300$ urban development projects, participation across three infrastructure domains—Sanitation ( $S$ ), Transport ( $T$ ), and Green Space ( $G$ )—was analyzed. The audit revealed that $145$ projects involve Sanitation, $135$ involve Transport, and $125$ involve Green Space. Exactly $70$ projects involve precisely two domains, while $20$ projects involve all three domains. Furthermore, $45$ projects involve both Sanitation and Transport, and $35$ projects involve both Transport and Green Space. Which of the following statements are correct? (Select all correct options.)

Exactly $70$ projects are dedicated exclusively to Sanitation.Answer
The total number of projects involving at least two domains is $90$ .Answer
The number of projects involving both Green Space and Sanitation, but not Transport, is $30$ .Answer
D
Exactly $15$ projects are not affiliated with any of the three domains.

Answer

The statements asserting that exactly 70 projects are dedicated exclusively to Sanitation, that 90 projects involve at least two domains, and that 30 projects involve both Green Space and Sanitation but not Transport are all correct.

The three correct options accurately reflect the regional values obtained from set decomposition: exclusive Sanitation projects equal 70, projects in two or more domains total 90 (70 + 20), and projects in Green Space and Sanitation only total 30.

Step-by-Step Solution

Calculate the regions involving exactly two domains

Sanitation and Transport only =

45 - 20 = 25

; Transport and Green Space only =

35 - 20 = 15

. Green Space and Sanitation only =

70 - (25 + 15) = 30

The intersection of two sets includes the triple intersection of all three sets.

Calculate single-domain exclusive regions

Sanitation only =

145 - (25 + 30 + 20) = 70

; Transport only =

135 - (25 + 15 + 20) = 75

; Green Space only =

125 - (30 + 15 + 20) = 60

Subtracting all overlapping two-set and three-set regions from each total set size isolates the single-domain counts.

Calculate projects in at least two domains

Projects in at least two domains = (Exactly 2 domains) + (All 3 domains) =

70 + 20 = 90

'At least two' is the union of the exactly two and exactly three regions.

Calculate total union and unaffiliated projects

Total in at least one domain =

70 + 75 + 60 + 25 + 15 + 30 + 20 = 295

. Unaffiliated projects =

300 - 295 = 5

Subtracting the union of all three sets from the total project pool yields the count of elements outside all sets.

Key Concept

Three-Set Venn Diagram Region Decomposition and Inclusion-Exclusion Principle

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