In a regional agricultural department, a survey was conducted among 160 field officers reg… — State PSC Exam

In a regional agricultural department, a survey was conducted among $160$ field officers regarding their supervisory duties across three major crop initiatives: Wheat ( $W$ ), Rice ( $R$ ), and Cotton ( $C$ ). The survey revealed that $85$ officers supervise Wheat, $70$ supervise Rice, and $60$ supervise Cotton. Furthermore, $35$ officers supervise both Wheat and Rice, $25$ supervise both Rice and Cotton, $30$ supervise both Wheat and Cotton, and $15$ officers supervise all three crop initiatives. How many field officers supervise exactly two of these three crop initiatives?

Answer: 45 officers

Answer

The total number of field officers who supervise exactly two crop initiatives is 45.

To find the number of officers supervising exactly two crop initiatives, we calculate the number of officers in each pair of overlapping sets exclusive of the three-set intersection: for Wheat and Rice only (

35 - 15 = 20

), for Rice and Cotton only (

25 - 15 = 10

), and for Wheat and Cotton only (

30 - 15 = 15

). Summing these three regions gives

20 + 10 + 15 = 45

Step-by-Step Solution

Extract intersection values for each pair of sets and subtract the central intersection of all three sets

Only Wheat and Rice =

35 - 15 = 20

; Only Rice and Cotton =

25 - 15 = 10

; Only Wheat and Cotton =

30 - 15 = 15

The given pairwise intersections include officers who supervise all three initiatives. To isolate those who supervise exactly two, the three-set intersection must be removed from each pairwise intersection.

Sum the three distinct 'exactly two' regions

Total =

20 + 10 + 15 = 45

Adding these disjoint counts yields the total number of officers supervising precisely two crop initiatives.

Key Concept

Inclusion-Exclusion Principle and Venn Diagram Region Analysis for Three Overlapping Sets

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