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Official PYQ · 2016Set 2 · QGA-06MCQ1 mark

GATE CS 2016 Set 2 QGA-06 — General Aptitude

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Among 150150 faculty members in an institute, 5555 are connected with each other through Facebook and 8585 are connected through Whatsapp. 3030 faculty members do not have Facebook or Whatsapp accounts. The numbers of faculty members connected only through Facebook accounts is _______.

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Answer: A

We can solve this problem using set theory or a Venn diagram.

Let:

  • UU be the total number of faculty members: ∣U∣=150|U| = 150
  • FF be the set of faculty members connected through Facebook: ∣F∣=55|F| = 55
  • WW be the set of faculty members connected through Whatsapp: ∣W∣=85|W| = 85

The number of faculty members who do not have either account is 3030. Therefore, the number of faculty members who have at least one of the accounts is:

∣F∪W∣=150−30=120|F \cup W| = 150 - 30 = 120

Using the principle of inclusion-exclusion:

∣F∪W∣=∣F∣+∣W∣−∣F∩W∣|F \cup W| = |F| + |W| - |F \cap W| 120=55+85−∣F∩W∣120 = 55 + 85 - |F \cap W| 120=140−∣F∩W∣120 = 140 - |F \cap W| ∣F∩W∣=20|F \cap W| = 20

Thus, the number of faculty members connected only through Facebook is:

∣F only∣=∣F∣−∣F∩W∣=55−20=35|F \text{ only}| = |F| - |F \cap W| = 55 - 20 = 35

Hence, the correct option is A.

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