In a survey of 120 civil service aspirants, 65 study Indian History, 55 study Indian Polit… — State PSC Exam

In a survey of $120$ civil service aspirants, $65$ study Indian History, $55$ study Indian Polity, and $45$ study Geography. Further details reveal that $25$ aspirants study both History and Polity, $20$ study both Polity and Geography, $15$ study both History and Geography, and $8$ study all three subjects. How many aspirants study exactly two subjects?

$36$ Cevap
B
$60$
C
$52$
D
$44$

Cevap

The number of aspirants studying exactly two subjects is 36.

The correct answer is

36

. To find the number of people in 'exactly two' sets in a 3-circle Venn diagram, we subtract the center region (all three sets) from each of the two-set overlapping regions:

(25 - 8) + (20 - 8) + (15 - 8) = 17 + 12 + 7 = 36

Adım Adım Çözüm

Identify the given region values from the problem statement

Total aspirants =

120

n(H) = 65

n(P) = 55

n(G) = 45

n(H \cap P) = 25

n(P \cap G) = 20

n(H \cap G) = 15

, and

n(H \cap P \cap G) = 8

Establishing set parameters enables standard Venn diagram region calculations.

Calculate aspirants studying only History and Polity

n(H \cap P \text{ only}) = n(H \cap P) - n(H \cap P \cap G) = 25 - 8 = 17

Sub-regions representing exactly two subjects must exclude elements in all three sets.

Calculate aspirants studying only Polity and Geography

n(P \cap G \text{ only}) = n(P \cap G) - n(H \cap P \cap G) = 20 - 8 = 12

Isolate the dual-subject overlap specific to Polity and Geography.

Calculate aspirants studying only History and Geography

n(H \cap G \text{ only}) = n(H \cap G) - n(H \cap P \cap G) = 15 - 8 = 7

Isolate the dual-subject overlap specific to History and Geography.

Sum the three exclusive dual-subject regions

17 + 12 + 7 = 36

The total number of aspirants studying exactly two subjects is the sum of the three mutually exclusive regions.

Anahtar Kavram

Inclusion-Exclusion Principle and Logical Venn Diagram Region Identification

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