In a administrative evaluation conducted for 250 government officers, data was collected r… — State PSC Exam

In a administrative evaluation conducted for $250$ government officers, data was collected regarding the completion of three specialized training modules: Cyber Security ( $C$ ), Data Analytics ( $D$ ), and Public Procurement ( $P$ ).

- $130$ officers completed Cyber Security
- $110$ officers completed Data Analytics
- $100$ officers completed Public Procurement
- $55$ officers completed both Cyber Security and Data Analytics
- $40$ officers completed both Data Analytics and Public Procurement
- $45$ officers completed both Cyber Security and Public Procurement
- $20$ officers completed all three modules

Based on the information provided, how many officers completed exactly two training modules?

$80$ Answer
B
$140$
C
$120$
D
$100$

Answer

The number of officers who completed exactly two training modules is

80

To find the number of officers who completed exactly two modules, we isolate the three disjoint regions that represent taking two modules but not the third. Subtraction of the three-module intersection (

20

) from each pair overlap yields:

(55 - 20) + (40 - 20) + (45 - 20) = 35 + 20 + 25 = 80

Step-by-Step Solution

Identify the given set values and intersections

N(C) = 130

N(D) = 110

N(P) = 100

N(C \cap D) = 55

N(D \cap P) = 40

N(C \cap P) = 45

N(C \cap D \cap P) = 20

Extracting all cardinalities provided in the stem.

Calculate the number of officers in each exclusive two-set intersection region

Cyber & Data only

= 55 - 20 = 35

; Data & Procurement only

= 40 - 20 = 20

; Cyber & Procurement only

= 45 - 20 = 25

The intersection

N(X \cap Y)

contains people taking all three modules as well. Subtracting

N(C \cap D \cap P)

isolates those taking ONLY two modules.

Sum the exclusive two-module regions

35 + 20 + 25 = 80

Adding the three disjoint regions corresponding to exactly two training modules.

Key Concept

3-Set Venn Diagram Region Isolation and Inclusion-Exclusion

Estimated Time:1m 30s

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