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Difficulty: HardOverlapping Sets and Venn Diagrams

In a survey of 200200 software developers regarding their proficiency in three programming languages—Python, Java, and C++:

- 110110 developers are proficient in Python.
- 9595 developers are proficient in Java.
- 8585 developers are proficient in C++.
- 2020 developers are proficient in all three languages.
- 3030 developers are proficient in none of the three languages.

If the total number of developers proficient in exactly two of these languages is twice the number of developers proficient in Python and Java only, how many developers are proficient in C++ only?

  1. A
    1515
  2. 2525Answer
  3. C
    3535
  4. D
    4040
  5. E
    4545

Answer

The number of developers proficient in C++ only is 25.
The correct answer is 25. Subtracting the 30 developers proficient in none of the languages leaves 170 developers who are proficient in at least one language. Using the 3-set overlap equations, the number of developers proficient in exactly two languages is calculated to be 80. Since the number proficient in Python and Java only is half of this (40), the sum of the remaining two double-overlap regions (Java & C++ only, and Python & C++ only) must also equal 40. Subtracting this combined double-overlap count of 40 and the triple-overlap count of 20 from the total C++ count of 85 leaves 25 developers proficient in C++ only.

Step-by-Step Solution

1
Calculate the total number of developers proficient in at least one language.
Total proficient=200−30=170\text{Total proficient} = 200 - 30 = 170
Developers proficient in none of the three languages must be excluded from the universe of set elements.
2
Apply the 3-set Venn Diagram formula relating total unique elements, individual set sums, exactly-two regions, and all-three regions.
Total=E1+E2+E3\text{Total} = E_1 + E_2 + E_3, where E1E_1 is exactly one, E2E_2 is exactly two, and E3E_3 is exactly three. Also, ∣P∣+∣J∣+∣C∣=E1+2E2+3E3|P| + |J| + |C| = E_1 + 2E_2 + 3E_3.

Sum of individual sets: 110+95+85=290110 + 95 + 85 = 290.
So, 290=E1+2E2+3(20)  ⟹  E1+2E2=230290 = E_1 + 2E_2 + 3(20) \implies E_1 + 2E_2 = 230.

From total elements: 170=E1+E2+20  ⟹  E1+E2=150170 = E_1 + E_2 + 20 \implies E_1 + E_2 = 150.

Subtracting the two equations gives E2=230−150=80E_2 = 230 - 150 = 80.
Each element in E2E_2 is counted twice in the sum of individual sets, and each element in E3E_3 is counted three times.
3
Determine the breakdown of the double-overlap regions.
Given that E2=2×(Python and Java only)E_2 = 2 \times (\text{Python and Java only}), we have Python and Java only=802=40\text{Python and Java only} = \frac{80}{2} = 40.

The remaining double-overlap regions involving C++ are (Java and C++ only)+(Python and C++ only)=E2−40=80−40=40(\text{Java and C++ only}) + (\text{Python and C++ only}) = E_2 - 40 = 80 - 40 = 40.
The total number of developers proficient in exactly two languages consists of three disjoint groups: Python/Java only, Java/C++ only, and Python/C++ only.
4
Calculate the number of developers proficient in C++ only.
∣C∣=(C++ only)+(Java and C++ only)+(Python and C++ only)+(All 3)85=(C++ only)+40+2085=(C++ only)+60C++ only=25|C| = (\text{C++ only}) + (\text{Java and C++ only}) + (\text{Python and C++ only}) + (\text{All 3}) 85 = (\text{C++ only}) + 40 + 20 85 = (\text{C++ only}) + 60 \text{C++ only} = 25
Subtract all overlapping subsets containing C++ from the total proficiency count of set C.

Key Concept

Three-Set Overlapping Venn Diagrams and Subset Decompositions
Estimated Time:2m 30s
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Topics

Overlapping Sets and Venn Diagrams

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