A survey was conducted among 300 State PSC aspirants regarding their preparation for three… — State PSC Exam

A survey was conducted among $300$ State PSC aspirants regarding their preparation for three subjects: General Studies ( $GS$ ), General Aptitude Test ( $C$ ), and Optional Subject ( $O$ ). The data collected is as follows:
- $180$ candidates prepare for $GS$
- $140$ candidates prepare for $C$
- $120$ candidates prepare for $O$
- $80$ candidates prepare for both $GS$ and $C$
- $50$ candidates prepare for both $C$ and $O$
- $60$ candidates prepare for both $GS$ and $O$
- $30$ candidates prepare for all three subjects

Which of the following statements are correct? (Select all correct statements)

The number of candidates preparing for exactly two subjects is $100$ .Answer
The number of candidates preparing for at least two subjects is $130$ .Answer
C
The number of candidates preparing for only General Studies ( $GS$ ) is $80$ .
The number of candidates who do not prepare for any of the three subjects is $20$ .Answer

Answer

The correct statements are those asserting that the number of candidates preparing for exactly two subjects is 100, the number of candidates preparing for at least two subjects is 130, and the number of candidates preparing for none of the three subjects is 20.

The statements asserting that exactly two subjects equal 100, at least two subjects equal 130, and none of the subjects equal 20 are all mathematically accurate based on set region decomposition: exactly two subjects count is

50 + 20 + 30 = 100

; at least two subjects is

100 + 30 = 130

; and outside all sets is

300 - 280 = 20

Step-by-Step Solution

Identify the central region (all three subjects).

The number of candidates preparing for all three subjects

n(GS \cap C \cap O) = 30

This value serves as the base subtraction term for all pairwise intersections.

Calculate the counts for candidates preparing for exactly two subjects.

GS and C only =

80 - 30 = 50

; C and O only =

50 - 30 = 20

; GS and O only =

60 - 30 = 30

. Total exactly two subjects =

50 + 20 + 30 = 100

Subtracting the triple intersection from each dual intersection isolates regions with exactly two subjects.

Calculate candidates preparing for only one subject.

Only GS =

180 - (50 + 30 + 30) = 70

; Only C =

140 - (50 + 30 + 20) = 40

; Only O =

120 - (30 + 30 + 20) = 40

Subtracting all overlapping regions from total set cardinalities yields single-subject counts.

Calculate the total union and the remainder outside all sets.

Total in at least one subject =

70 + 40 + 40 + 50 + 20 + 30 + 30 = 280

. Neither subject =

300 - 280 = 20

Applying inclusion-exclusion principle determines the complete universe coverage.

Key Concept

Three-Set Principle of Inclusion-Exclusion

Estimated Time:2m 0s

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