GATE EE 2016 Set 1 — Question 6

Let $N$ be the total number of faculty members, $N = 150$.
Let $F$ be the set of faculty members with Facebook accounts. Given $|F| = 55$.
Let $W$ be the set of faculty members with WhatsApp accounts. Given $|W| = 85$.We are given that 30 faculty members do not have Facebook® or WhatsApp® accounts. This means they are neither in $F$ nor in $W$.The number of faculty members who have at least one account (Facebook or WhatsApp or both) is:
$|F \cup W| = N - (\text{number of members with neither account})$
$|F \cup W| = 150 - 30 = 120$.Using the Principle of Inclusion-Exclusion for two sets:
$|F \cup W| = |F| + |W| - |F \cap W|$Substitute the known values:
$120 = 55 + 85 - |F \cap W|$
$120 = 140 - |F \cap W|$Now, solve for $|F \cap W|$, which represents the number of faculty members who have both Facebook and WhatsApp accounts:
$|F \cap W| = 140 - 120 = 20$.The question asks for the number of faculty members connected only through Facebook accounts.
This can be calculated as the total number of Facebook users minus those who also use WhatsApp:
$(\text{Only Facebook}) = |F| - |F \cap W|$
$(\text{Only Facebook}) = 55 - 20 = 35$.Thus, 35 faculty members are connected only through Facebook accounts.The final answer is $\boxed{A}$