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Difficulty: MediumOverlapping Sets and Venn Diagrams

A technology company conducted a survey of its 200200 software engineers regarding their proficiency in three programming languages: Python, Java, and C++. The survey revealed the following data:
- 110110 engineers are proficient in Python.
- 8585 engineers are proficient in Java.
- 7575 engineers are proficient in C++.
- 4040 engineers are proficient in both Python and Java.
- 3535 engineers are proficient in both Java and C++.
- 3030 engineers are proficient in both Python and C++.
- 1515 engineers are proficient in all three programming languages.

How many of the surveyed engineers are proficient in none of these three programming languages?

  1. A
    15
  2. 20Answer
  3. C
    25
  4. D
    35
  5. E
    50

Answer

The number of software engineers proficient in none of the three programming languages is 20.
Using the 3-set inclusion-exclusion formula, the total number of engineers proficient in at least one programming language is 110+85+75−(40+35+30)+15=180110 + 85 + 75 - (40 + 35 + 30) + 15 = 180. Subtracting this from the total sample size of 200200 yields 200−180=20200 - 180 = 20 engineers proficient in none of the three languages.

Step-by-Step Solution

1
State the Principle of Inclusion-Exclusion formula for three sets.
N(At least one)=N(P)+N(J)+N(C)−[N(P∩J)+N(J∩C)+N(P∩C)]+N(P∩J∩C)N(\text{At least one}) = N(P) + N(J) + N(C) - [N(P \cap J) + N(J \cap C) + N(P \cap C)] + N(P \cap J \cap C)
To find the total number of engineers in at least one category without double-counting overlapping groups.
2
Substitute the given values into the formula.
N(At least one)=110+85+75−(40+35+30)+15=270−105+15=180N(\text{At least one}) = 110 + 85 + 75 - (40 + 35 + 30) + 15 = 270 - 105 + 15 = 180
Calculate the total number of unique engineers who know at least one language.
3
Subtract the number of engineers proficient in at least one language from the total population.
N(None)=200−180=20N(\text{None}) = 200 - 180 = 20
The total population consists of engineers proficient in at least one language plus those proficient in none.

Key Concept

Three-Set Overlapping Venn Diagrams and the Principle of Inclusion-Exclusion
Estimated Time:2m 0s
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Topics

Overlapping Sets and Venn Diagrams

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