In a district administrative audit of 240 agricultural cooperatives, compliance with three… — State PSC Exam

In a district administrative audit of $240$ agricultural cooperatives, compliance with three digital platforms was surveyed: E-Farming Portal ( $P$ ), Soil Health Database ( $S$ ), and Crop Insurance Registry ( $I$ ). It was found that $115$ cooperatives use $P$ , $110$ use $S$ , and $105$ use $I$ . Exactly $25$ cooperatives do not use any of the three platforms, and $20$ cooperatives use all three platforms. Furthermore, the number of cooperatives using only $P$ , only $S$ , and only $I$ are in the ratio $8 : 9 : 7$ , respectively. How many cooperatives use exactly two of these digital platforms?

75Answer
B
95
C
55
D
90

Answer

The number of cooperatives using exactly two digital platforms is 75.

By applying the principle of inclusion-exclusion across the 3 sets, the union of all three platforms is

240 - 25 = 215

. Expressing the total elements as the sum of single-platform users

(a+b+c)

, exactly-two-platform users

(d+e+f)

, and all-three-platform users (

20

), we get

(a+b+c) + (d+e+f) = 195

. Summing the individual set totals without the central intersection yields

(a+b+c) + 2(d+e+f) = 95 + 90 + 85 = 270

. Subtracting the first equation from the second eliminates

(a+b+c)

directly, leaving

d+e+f = 270 - 195 = 75

Step-by-Step Solution

Calculate the total number of cooperatives using at least one digital platform.

Total using at least one platform =

240 - 25 = 215

Subtracting the cooperatives that use none of the platforms from the total audited population gives the union of all three sets

|P \cup S \cup I|

Set up equations for single-region and double-region intersection counts using set partition variables.

Let

a, b, c

be the counts for only

P

, only

S

, and only

I

respectively. Let

d, e, f

be the counts for exactly two platforms (

P \cap S

only,

S \cap I

only,

P \cap I

only). The count for all three platforms is

g = 20

Dividing the 3-set Venn diagram into 7 mutually exclusive regions allows exact algebraic formulation.

Express total set union and individual set totals in terms of these regions.

Equation 1 (Union):

(a + b + c) + (d + e + f) + 20 = 215 \implies (a + b + c) + (d + e + f) = 195

.
Individual sets:

|P| = a + d + f + 20 = 115 \implies a + d + f = 95

|S| = b + d + e + 20 = 110 \implies b + d + e = 90

|I| = c + e + f + 20 = 105 \implies c + e + f = 85

Each individual set sum accounts for its unique region, adjacent two-set intersections, and the three-set intersection.

Sum the three individual set equations and solve for

(d + e + f)

Equation 2 (Sum of set equations):

(a + b + c) + 2(d + e + f) = 95 + 90 + 85 = 270

.
Subtracting Equation 1 from Equation 2:

[(a + b + c) + 2(d + e + f)] - [(a + b + c) + (d + e + f)] = 270 - 195

(d + e + f) = 75

Subtracting the union sum eliminates the single-region terms

(a + b + c)

directly, isolating the sum of regions representing exactly two platforms.

Key Concept

Principle of Inclusion-Exclusion and Region Partitioning in 3-Set Venn Diagrams

Estimated Time:3m 0s

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