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Difficulty: EasyVenn Diagrams and Set-Based Data

In a civil services training academy of 120120 officers, 7070 officers speak English, 6060 officers speak Hindi, and 3030 officers speak both English and Hindi. Based on this set data, which of the following statements are correct?

  1. The number of officers who speak at least one of the two languages is 100100.Answer
  2. The number of officers who speak only English is 4040.Answer
  3. C
    The number of officers who speak neither language is 3030.
  4. The number of officers who speak exactly one language is 7070.Answer

Answer

The correct statements are: the number of officers who speak at least one of the two languages is 100100; the number of officers who speak only English is 4040; and the number of officers who speak exactly one language is 7070.
Statements asserting that 100100 officers speak at least one language, 4040 speak only English, and 7070 speak exactly one language are mathematically correct based on standard set operations: ∣E∪H∣=70+60−30=100|E \cup H| = 70 + 60 - 30 = 100, Only E=70−30=40E = 70 - 30 = 40, and Exactly One =40+30=70= 40 + 30 = 70.

Step-by-Step Solution

1
Calculate the number of officers speaking at least one language using Inclusion-Exclusion Principle.
∣E∪H∣=∣E∣+∣H∣−∣E∩H∣=70+60−30=100|E \cup H| = |E| + |H| - |E \cap H| = 70 + 60 - 30 = 100.
The intersection must be subtracted once to avoid double counting.
2
Find the number of officers speaking only English and only Hindi.
Only English = 70−30=4070 - 30 = 40; Only Hindi = 60−30=3060 - 30 = 30.
Subtracting the overlap from each set leaves elements exclusive to that set.
3
Calculate officers speaking neither language and officers speaking exactly one language.
Neither = 120−100=20120 - 100 = 20; Exactly one = 40+30=7040 + 30 = 70.
Neither language is the complement of the union. Exactly one language is the sum of disjoint set regions.

Key Concept

Two-set inclusion-exclusion principle and disjoint region classification in set theory
Estimated Time:50s
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Topics

Venn Diagrams and Set-Based Data

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