In a state administration academy, a cohort of 150 officer trainees completed specialized … — State PSC Exam

In a state administration academy, a cohort of $150$ officer trainees completed specialized skill modules in Artificial Intelligence ( $A$ ), Blockchain Technology ( $B$ ), and Cyber Security ( $C$ ). Data from the academy administration indicates:
- $75$ trainees enrolled in Artificial Intelligence ( $A$ )
- $65$ trainees enrolled in Blockchain Technology ( $B$ )
- $60$ trainees enrolled in Cyber Security ( $C$ )
- $30$ trainees enrolled in both Artificial Intelligence and Blockchain Technology
- $25$ trainees enrolled in both Blockchain Technology and Cyber Security
- $20$ trainees enrolled in both Artificial Intelligence and Cyber Security
- $10$ trainees enrolled in all three modules

Based on the provided data, which of the following statements regarding the trainee enrollment are correct?

The total number of trainees enrolled in exactly one training module is $80$ .Answer
B
The total number of trainees enrolled in exactly two training modules is $75$ .
The total number of trainees enrolled in at least two training modules is $55$ .Answer
D
The total number of trainees enrolled in none of the three training modules is $25$ .

Answer

The statements asserting that

80

trainees are enrolled in exactly one training module and that

55

trainees are enrolled in at least two training modules are correct.

The correct evaluation shows that exactly one module enrollment totals

35 + 20 + 25 = 80

, and at least two modules enrollment totals

45 + 10 = 55

Step-by-Step Solution

Identify the region for trainees taking all three modules.

n(A \cap B \cap C) = 10

This fundamental triple intersection value must be subtracted from pairwise intersections to isolate dual-module regions.

Calculate trainees enrolled in exactly two modules.

Only

A \text{ and } B = 30 - 10 = 20

; Only

B \text{ and } C = 25 - 10 = 15

; Only

A \text{ and } C = 20 - 10 = 10

. Total taking exactly two modules =

20 + 15 + 10 = 45

Subtracting the triple intersection isolates individuals taking exclusively two subjects.

Calculate trainees enrolled in exactly one module.

Only

A = 75 - (20 + 10 + 10) = 35

; Only

B = 65 - (20 + 15 + 10) = 20

; Only

C = 60 - (10 + 15 + 10) = 25

. Total taking exactly one module =

35 + 20 + 25 = 80

Subtracting dual and triple overlaps from total single set numbers yields single-module counts.

Determine total trainees taking at least one module and trainees taking none.

Total taking at least one module

= 80 \text{ (single)} + 45 \text{ (double)} + 10 \text{ (triple)} = 135

. Trainees taking none

= 150 - 135 = 15

Applying set summation gives total enrolled trainees, from which the non-enrolled count is derived.

Key Concept

Three-Set Venn Diagram Region Decomposition using Inclusion-Exclusion Principle

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic