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For the output $\text{F}$ to be $1$ in the logic circuit shown, the input combination should be

  1. $\mathrm{A}=1, \mathrm{~B}=1, \mathrm{C}=0$
  2. $\text{A = 1, B = 0, C = 0}$
  3. $\mathrm{A}=0, \mathrm{~B}=1, \mathrm{C}=0$
  4. $\text{A = 0, B = 0, C = 1}$
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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 $4.$

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 

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