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Difficulty: EasyOverlapping Sets and Inclusion-Exclusion Principle

In a technology firm of 100100 software engineers, 6565 engineers know Python and 4545 know Java. If 2020 engineers know both Python and Java, how many of the engineers know neither Python nor Java?

  1. 1010Answer
  2. B
    00
  3. C
    1515
  4. D
    2020
  5. E
    3030

Answer

The number of engineers who know neither Python nor Java is 1010.
The total number of engineers is 100100. By inclusion-exclusion, the union of Python and Java speakers is 65+45−20=9065 + 45 - 20 = 90. The number of engineers who know neither language is the total minus the union, 100−90=10100 - 90 = 10.

Step-by-Step Solution

1
Calculate the number of engineers who know at least one language using the Inclusion-Exclusion Principle.
At least one=Python+Java−Both=65+45−20=90\text{At least one} = \text{Python} + \text{Java} - \text{Both} = 65 + 45 - 20 = 90
Engineers who know both languages are counted in both individual totals, so their count must be subtracted once to avoid double counting.
2
Subtract the number of engineers who know at least one language from the total number of engineers to find those who know neither.
Neither=Total−At least one=100−90=10\text{Neither} = \text{Total} - \text{At least one} = 100 - 90 = 10
The total group consists of those who know at least one language plus those who know neither.

Key Concept

Two-Set Inclusion-Exclusion Principle
Estimated Time:45s
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Topics

Overlapping Sets and Inclusion-Exclusion Principle

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