In a state revenue department consisting of 200 inspectors, assignments are carried out ac… — State PSC Exam

In a state revenue department consisting of $200$ inspectors, assignments are carried out across three specialized sectors: Taxation ( $T$ ), Customs ( $C$ ), and Financial Intelligence ( $F$ ). A departmental survey revealed that $105$ inspectors handle Taxation, $90$ handle Customs, and $85$ handle Financial Intelligence. Furthermore, $45$ inspectors handle both Taxation and Customs, $40$ handle both Customs and Financial Intelligence, and $35$ handle both Taxation and Financial Intelligence. If $15$ inspectors do not work in any of these three sectors, how many inspectors are engaged in EXACTLY TWO of the three sectors?

45Answer
B
70
C
95
D
120

Answer

45 inspectors are engaged in exactly two of the three sectors.

The option stating 45 is correct. By using the principle of inclusion-exclusion, the total number of inspectors operating in at least one sector is

200 - 15 = 185

. Solving

185 = (105 + 90 + 85) - (45 + 40 + 35) + |T \cap C \cap F|

yields

|T \cap C \cap F| = 25

. To find those in exactly two sectors, we subtract 25 from each pairwise intersection (

45-25=20

40-25=15

35-25=10

), yielding a total of

20 + 15 + 10 = 45

Step-by-Step Solution

Determine the total number of inspectors working in at least one sector

|T \cup C \cup F| = 200 - 15 = 185

Inspectors who work in at least one sector equal the total inspectors minus those in none of the sectors.

Apply the Principle of Inclusion-Exclusion to find the number of inspectors in all three sectors

|T \cup C \cup F| = |T| + |C| + |F| - (|T \cap C| + |C \cap F| + |T \cap F|) + |T \cap C \cap F| \Rightarrow 185 = 105 + 90 + 85 - (45 + 40 + 35) + |T \cap C \cap F| \Rightarrow 185 = 280 - 120 + |T \cap C \cap F| \Rightarrow |T \cap C \cap F| = 25

The standard set identity relates the union of three sets with their individual sizes and overlapping regions.

Calculate the inspectors working in EXACTLY two sectors for each pairwise combination

Taxation and Customs only:

45 - 25 = 20

; Customs and Financial Intelligence only:

40 - 25 = 15

; Taxation and Financial Intelligence only:

35 - 25 = 10

Inspectors in all three sectors are included in every pairwise intersection and must be subtracted to isolate those in exactly two sectors.

Sum the counts of inspectors in exactly two sectors

20 + 15 + 10 = 45

Adding these disjoint sets gives the total count of inspectors working in exactly two sectors.

Key Concept

Principle of Inclusion-Exclusion for 3 Sets

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic