A survey was conducted among 400 administrative officers in a state secretariat regarding … — State PSC Exam

A survey was conducted among $400$ administrative officers in a state secretariat regarding their operational oversight of three major public welfare projects: Project Alpha, Project Beta, and Project Gamma. The survey collected the following data:

- $190$ officers oversee Project Alpha.
- $180$ officers oversee Project Beta.
- $170$ officers oversee Project Gamma.
- $85$ officers oversee both Project Alpha and Project Beta.
- $75$ officers oversee both Project Beta and Project Gamma.
- $70$ officers oversee both Project Alpha and Project Gamma.
- $40$ officers do not oversee any of these three projects.

Based on the given information, how many administrative officers oversee exactly two projects?

$80$ Answer
B
$130$
C
$180$
D
$230$

Answer

The number of administrative officers who oversee exactly two projects is

80

The value

80

is correct because subtracting those in no projects from the total of

400

gives

360

officers in at least one project. Using the inclusion-exclusion principle formula yields

50

officers in all three projects. Subtracting

50

from each pairwise intersection gives

35

25

, and

20

for the three exclusive two-project regions, summing to

80

Step-by-Step Solution

Calculate the total number of officers supervising at least one project.

|A \cup B \cup C| = 400 - 40 = 360

Officers who oversee at least one project are found by subtracting those overseeing none from the total surveyed officers.

Apply the Principle of Inclusion-Exclusion for three sets to find the triple intersection (

|A \cap B \cap C|

360 = 190 + 180 + 170 - (85 + 75 + 70) + |A \cap B \cap C| \implies 360 = 540 - 230 + |A \cap B \cap C| \implies |A \cap B \cap C| = 50

The standard set formula accounts for single set totals, pairwise intersections, and the three-set intersection.

Determine the number of officers in each exclusive two-project intersection.

Only Alpha & Beta =

85 - 50 = 35

; Only Beta & Gamma =

75 - 50 = 25

; Only Alpha & Gamma =

70 - 50 = 20

Subtracting the triple intersection from each pairwise intersection yields the count of officers managing exactly two projects.

Sum the exclusive two-project regions.

35 + 25 + 20 = 80

Adding these three disjoint set regions gives the total number of officers handling exactly two projects.

Key Concept

Three-Set Inclusion-Exclusion Principle and Set Region Decomposition

Estimated Time:2m 30s

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