A technology firm surveyed 100 software developers regarding their proficiency in three pr… — GRE General Test

A technology firm surveyed $100$ software developers regarding their proficiency in three programming languages: Python ( $P$ ), Java ( $J$ ), and C++ ( $C$ ). The survey revealed the following results:
- $55$ developers are proficient in Python.
- $50$ developers are proficient in Java.
- $35$ developers are proficient in C++.
- $25$ developers are proficient in both Python and Java.
- $20$ developers are proficient in both Java and C++.
- $15$ developers are proficient in both Python and C++.
- $10$ developers are proficient in none of these three languages.

Which of the following statements must be true? Select all such statements.

Exactly $10$ developers are proficient in all three programming languages.Cevap
B
Exactly $50$ developers are proficient in at least two of these programming languages.
The number of developers proficient in Python only is $25$ .Cevap
The total number of developers proficient in exactly one of these languages is $50$ .Cevap
E
The percentage of Python-proficient developers who are proficient in Python only is $25\%$ .

Cevap

The correct statements are those asserting that exactly 10 developers know all three languages, that 25 developers know Python only, and that 50 developers know exactly one of these languages.

The statements confirming that 10 developers know all three languages, 25 know Python only, and 50 know exactly one language are correct based on standard 3-set Venn diagram calculations. Using Inclusion-Exclusion,

|P \cup J \cup C| = 90

, which gives

|P \cap J \cap C| = 10

. Subtracting overlapping regions yields

25

for Python only,

15

for Java only,

10

for C++ only, summing to

50

for exactly one language.

Adım Adım Çözüm

Calculate the total number of developers proficient in at least one language.

N_{\text{at least one}} = 100 - 10 = 90

Subtracting the developers proficient in none of the languages from the total survey population yields the union

|P \cup J \cup C|

Apply the Principle of Inclusion-Exclusion for 3 sets to find the triple intersection

|P \cap J \cap C|

90 = 55 + 50 + 35 - (25 + 20 + 15) + |P \cap J \cap C| \implies 90 = 80 + |P \cap J \cap C| \implies |P \cap J \cap C| = 10

The formula

|P \cup J \cup C| = |P| + |J| + |C| - (|P \cap J| + |J \cap C| + |P \cap C|) + |P \cap J \cap C|

links all given set quantities.

Calculate the count of developers proficient in exactly two languages for each pair.

Python and Java only =

25 - 10 = 15

; Java and C++ only =

20 - 10 = 10

; Python and C++ only =

15 - 10 = 5

Subtracting the triple intersection count (

10

) from each pairwise intersection yields the exclusive two-set regions.

Calculate the single-language proficiency counts (exactly one language).

Python only =

55 - (15 + 5 + 10) = 25

; Java only =

50 - (15 + 10 + 10) = 15

; C++ only =

35 - (5 + 10 + 10) = 10

. Total exactly one =

25 + 15 + 10 = 50

Subtracting all overlapping regions containing each language from its total count yields the single-language region size.

Evaluate the statement options based on computed regional values.

Statements stating 10 all three, 25 Python only, and 50 exactly one language are true.

Comparing calculated values (

10

for all three,

25

for Python only,

50

for exactly one language) against each option validates the true choices.

Anahtar Kavram

Three-set inclusion-exclusion principle and Venn diagram region decomposition

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