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Difficulty: EasyLogical Venn Diagrams

In a group of 6060 civil service aspirants, 3535 aspirants study History, 2525 study Geography, and 1010 study both History and Geography. How many aspirants study neither History nor Geography?

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

Answer

The number of aspirants studying neither History nor Geography is 1010.
By applying the set formula ∣H∪G∣=∣H∣+∣G∣−∣H∩G∣|H \cup G| = |H| + |G| - |H \cap G|, we get ∣H∪G∣=35+25−10=50|H \cup G| = 35 + 25 - 10 = 50. The number of aspirants who study neither subject is 60−50=1060 - 50 = 10.

Step-by-Step Solution

1
Calculate the total number of aspirants studying at least one subject using the principle of inclusion-exclusion.
Aspirants studying at least one subject = 35+25−10=5035 + 25 - 10 = 50.
The intersection of 1010 is counted in both individual groups and must be subtracted once to avoid double counting.
2
Subtract the number of aspirants studying at least one subject from the total population.
Aspirants studying neither subject = 60−50=1060 - 50 = 10.
The total group consists of aspirants studying at least one subject plus those studying neither.

Key Concept

Principle of Inclusion-Exclusion for two sets: ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|.
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Logical Venn Diagrams

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