A technology institute surveyed a cohort of 300 software engineers regarding their profici… — GRE General Test

A technology institute surveyed a cohort of $300$ software engineers regarding their proficiency in three programming paradigms: Functional ( $F$ ), Object-Oriented ( $O$ ), and Concurrent ( $C$ ). Every surveyed engineer is proficient in at least one of these three paradigms. The ratio of the total number of engineers proficient in $F$ , $O$ , and $C$ is $5 : 6 : 4$ , respectively. Furthermore, exactly $20\%$ of the engineers proficient in $F$ are proficient in all three paradigms. If exactly $54$ engineers are proficient in both $F$ and $O$ , $48$ are proficient in both $O$ and $C$ , and $30$ are proficient in both $F$ and $C$ , how many engineers in the cohort are proficient in exactly one programming paradigm?

A
96
B
141
222Answer
D
273
E
78

Answer

222 engineers are proficient in exactly one programming paradigm.

Using the Principle of Inclusion-Exclusion for three sets,

|F \cup O \cup C| = |F| + |O| + |C| - (|F \cap O| + |O \cap C| + |F \cap C|) + |F \cap O \cap C|

. Substituting

|F|=5k

|O|=6k

|C|=4k

|F \cap O \cap C|=k

, and the total cohort

300

gives

300 = 16k - 132

, yielding

k = 27

. Consequently, the triple intersection is

27

. Decomposing into disjoint regions:

F

only

= 78

O

only

= 87

, and

C

only

= 57

. Summing these gives

78 + 87 + 57 = 222

Step-by-Step Solution

Set up algebraic representations for the set sizes using the given ratio.

Let

|F| = 5k

|O| = 6k

, and

|C| = 4k

for some positive constant

k

The total proficiencies follow the ratio

5:6:4

Express the triple intersection

|F \cap O \cap C|

in terms of

k

|F \cap O \cap C| = 0.20 \times |F| = 0.20 \times 5k = k

Exactly 20% of engineers proficient in

F

are proficient in all three paradigms.

Apply the Principle of Inclusion-Exclusion for three sets to solve for

k

|F \cup O \cup C| = |F| + |O| + |C| - (|F \cap O| + |O \cap C| + |F \cap C|) + |F \cap O \cap C| \implies 300 = 5k + 6k + 4k - (54 + 48 + 30) + k \implies 300 = 16k - 132 \implies 16k = 432 \implies k = 27

Every engineer is proficient in at least one paradigm, so

|F \cup O \cap C| = 300

Calculate the total size of each set and each exclusive intersection region.

|F| = 135

|O| = 162

|C| = 108

, and

|F \cap O \cap C| = 27

.
Exclusively

F \cap O = 54 - 27 = 27

.
Exclusively

O \cap C = 48 - 27 = 21

.
Exclusively

F \cap C = 30 - 27 = 3

Subtracting the triple intersection from pairwise intersections yields the two-set-only regions.

Determine the number of engineers proficient in exactly one paradigm.

Only

F = 135 - (27 + 3 + 27) = 78

.
Only

O = 162 - (27 + 21 + 27) = 87

.
Only

C = 108 - (3 + 21 + 27) = 57

.
Total exactly one =

78 + 87 + 57 = 222

Subtracting all overlapping regions from each total set size gives the single-category populations.

Key Concept

Three-set Principle of Inclusion-Exclusion and Venn Diagram region decomposition.

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