GATE CS 2016 Set 2 Q.6 (GA Q.6): Quantitative Aptitude

Among 150 faculty members in an institute, 55 are connected with each other through Facebook and 85 are connected through WhatsApp. 30 faculty members do not have Facebook or WhatsApp accounts. The number of faculty members connected only through Facebook accounts is ______________.

Official answer and solution

Official answer: A
From: Official GATE 2016 answer key

Solution

Apply the Principle of Inclusion-Exclusion on the two overlapping sets of social media users.

Let $U$ denote the set of all faculty members in the institute. We are given:

|U| = 150

Let $F$ be the set of members on Facebook, and $W$ be the set of members on WhatsApp:

$|F| = 55$
$|W| = 85$

There are 30 members with neither account, meaning:

|U \setminus (F \cup W)| = 30 \implies |F \cup W| = 150 - 30 = 120

Using the Principle of Inclusion-Exclusion:

|F \cup W| = |F| + |W| - |F \cap W|

120 = 55 + 85 - |F \cap W|

120 = 140 - |F \cap W| \implies |F \cap W| = 140 - 120 = 20

The number of faculty members connected ONLY through Facebook is:

|F \setminus W| = |F| - |F \cap W| = 55 - 20 = 35

Therefore, 35 faculty members are connected only through Facebook.

Final answer

Common trap

Subtracting $|F \cap W|$ from total members instead of from $|F|$ , or forgetting to subtract the 30 non-users first.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2016 Computer Science and Information Technology (CS) Session 2
Question: 6 of 65, printed as GA Q.6
Section: General Aptitude

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