Detailed Video Solution, with Complete Analysis: https://youtu.be/edSHGdnHBdw
Let $X = A \oplus B$ ; So, $\overline{X} = A \odot B$
The output $F = \overline{X \oplus \overline{X} \oplus C }$
So, $F = C$
Note that the two of the inputs of the final XNor gate are always opposite($X$, $\overline{X}$), hence, $\mathrm{F} = C.$
Hence, for $F$ to be 1; Inputs $\text{A, B}$ can be anything, But $\text{C}$ must be $1.$
So, answer is Option D.
& The number of input combinations $\text{(A, B, C)}$ for which the output $\text{F}$ becomes $1$ is $4.$
Detailed Video Solution, with Complete Analysis: https://youtu.be/edSHGdnHBdw
The final gate in the given circuit is $XNOR-3$ gate i.e. $XNOR$ gate with $3$ inputs.
A Very Important NOTE is:
$XNOR-3$ gate with inputs $A,B,C$ is NOT same as $A \odot B \odot C .$
$XNOR-n$ gate is defined as Complement of $XOR-n$ gate.
NOTE that:
$a \odot b \odot c = a \oplus b \oplus c$
BUT $XNOR-3 \,\, gate \neq XOR-3 \,\, gate$
The exclusive‐NOR gate is the complement of the exclusive‐OR gate, as indicated by the small circle on the output side of the graphic symbol.
3 Inputs XNOR gate, Complete Analysis: https://youtu.be/uAadNn38oFo
XOR & XNOR functions: https://www.youtube.com/watch?v=-30dUjh6Qv4
After watching THIS video solution, Solve this GATE EC 2015 question: https://ec.gateoverflow.in/631/gate-ece-2015-set-1-question-38