In a survey of 120 civil service aspirants, 65 read Newspaper A, 55 read Newspaper B, and … — State PSC Exam

In a survey of $120$ civil service aspirants, $65$ read Newspaper A, $55$ read Newspaper B, and $45$ read Newspaper C. Additionally, $25$ read both A and B, $20$ read both B and C, $15$ read both A and C, and $8$ read all three newspapers. How many aspirants read exactly two of these newspapers?

36Answer
B
60
C
52
D
44

Answer

36 aspirants read exactly two newspapers.

To find the number of aspirants who read exactly two newspapers, we must subtract the number of aspirants who read all three newspapers (8) from each of the two-newspaper intersection groups. The number of aspirants reading only A and B is

25 - 8 = 17

, only B and C is

20 - 8 = 12

, and only A and C is

15 - 8 = 7

. Summing these exclusive regions gives

17 + 12 + 7 = 36

Step-by-Step Solution

Identify the given set values and intersections

Total aspirants

N = 120

;

n(A \cap B) = 25

;

n(B \cap C) = 20

;

n(A \cap C) = 15

;

n(A \cap B \cap C) = 8

We need to extract overlapping region counts to isolate the 'exactly two' regions.

Calculate aspirants reading ONLY two newspapers for each pair

Only A and B =

25 - 8 = 17

; Only B and C =

20 - 8 = 12

; Only A and C =

15 - 8 = 7

The given pairwise intersections

n(A \cap B)

include those who read all three newspapers, so

n(A \cap B \cap C)

must be removed from each pair.

Sum the exclusive two-set regions

17 + 12 + 7 = 36

Adding these three mutually exclusive regions yields the total number of aspirants reading exactly two newspapers.

Key Concept

Venn Diagram set decomposition and region isolation

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