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EX-OR gate is the odd number of 1's detector whereas EX-NOR gate is even number of 1's detector.

EX-OR gate gives output 1 if a number of 1's in the input is odd.

EX-NOR gate gives output 1 if a number of 1's in the input is even.

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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 $4.$


NOTE:

3-input XOR function is SAME as 3-input XNOR function.

BUT

3-input XNOR Gate is NOT same 3-input XOR Gate.

Watch this: XNOR Gate Vs XNOR Function | 3 Inputs XNOR gate | GATE EC 2010, GATE EC 2015


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   

Answer:

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