In an agricultural extension survey conducted among 250 farmers in an administrative block… — State PSC Exam

In an agricultural extension survey conducted among $250$ farmers in an administrative block, awareness of three welfare schemes was evaluated: Crop Insurance ( $C$ ), Soil Health Card ( $S$ ), and Kisan Credit Card ( $K$ ). The survey revealed that $120$ farmers are aware of $C$ , $110$ are aware of $S$ , and $130$ are aware of $K$ . Additionally, $45$ farmers are aware of both $C$ and $S$ , $50$ are aware of both $S$ and $K$ , and $40$ are aware of both $C$ and $K$ . If $15$ farmers are aware of all three schemes, how many farmers are aware of exactly two of these schemes?

Answer: 90 farmers

Answer

The total number of farmers aware of exactly two schemes is

90

To find the number of farmers aware of exactly two schemes, we isolate the three two-set intersection regions that exclude the three-set intersection. Subtracting the

15

farmers aware of all three schemes from each pairwise intersection yields

30

(for

C

and

S

only),

35

(for

S

and

K

only), and

25

(for

C

and

K

only). Summing these mutually exclusive regions gives

30 + 35 + 25 = 90

Step-by-Step Solution

Calculate the count of farmers aware of only Crop Insurance and Soil Health Card

45 - 15 = 30

The pairwise intersection includes farmers aware of all three schemes, so subtracting the triple intersection isolates those aware of only these two schemes.

Calculate the count of farmers aware of only Soil Health Card and Kisan Credit Card

50 - 15 = 35

Subtract the triple intersection count from the pairwise intersection count of

S

and

K

Calculate the count of farmers aware of only Crop Insurance and Kisan Credit Card

40 - 15 = 25

Subtract the triple intersection count from the pairwise intersection count of

C

and

K

Sum the three region counts for exactly two schemes

30 + 35 + 25 = 90

The regions representing 'only C and S', 'only S and K', and 'only C and K' are mutually exclusive.

Key Concept

3-Set Venn Diagram Region Isolation

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