In a survey of 180 cloud computing engineers regarding their proficiency in three major pl… — JAMB UTME

In a survey of $180$ cloud computing engineers regarding their proficiency in three major platforms—AWS ( $A$ ), Azure ( $B$ ), and Google Cloud ( $C$ )—it was found that $95$ are proficient in AWS, $80$ in Azure, and $75$ in Google Cloud. Furthermore, $40$ are proficient in both AWS and Azure, $35$ in both Azure and Google Cloud, $30$ in both AWS and Google Cloud, and $15$ are not proficient in any of the three platforms. How many engineers are proficient in exactly one of these platforms?

Answer: 100 engineers

Answer

100 engineers are proficient in exactly one platform.

Using the 3-set inclusion-exclusion formula, the number of engineers proficient in all three platforms is solved as 20. Subtracting the relevant overlap regions from each set gives 45 for AWS only, 25 for Azure only, and 30 for Google Cloud only. Adding these single-set values yields a total of 100 engineers proficient in exactly one platform.

Step-by-Step Solution

Determine the cardinality of the union of all three sets.

n(A \cup B \cup C) = 180 - 15 = 165

Subtracting the engineers who are not proficient in any of the three platforms from the universal set.

Apply the Principle of Inclusion-Exclusion to calculate the triple intersection

n(A \cap B \cap C)

n(A \cap B \cap C) = 20

Substituting known values gives

165 = 250 - 105 + n(A \cap B \cap C)

, which simplifies to

n(A \cap B \cap C) = 20

Calculate the counts for regions representing exactly two platforms.

AWS & Azure only = 20, Azure & GCP only = 15, AWS & GCP only = 10.

Subtracting the triple intersection count (20) from each pairwise intersection.

Calculate the single-set exclusive regions.

AWS only = 45, Azure only = 25, GCP only = 30.

Subtracting all multi-platform overlap regions from each total set size.

Sum the single-set exclusive regions.

Total =

45 + 25 + 30 = 100

Combining the counts of engineers proficient in AWS only, Azure only, and Google Cloud only.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Venn Diagram Region Partitioning

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