What is the gain in a non-inverting amplifier
Here is my interpretation of the OP's question and explanations (I have considered the simple case of connecting to ground):

&quot;The input signal is connected to terminal E4&quot; means &quot;The input signal is directly connected to terminal E4&quot;. So R3, Rref, Vref, E3 and E5 (if only not connected to ground) do not play any role in forming the voltage of the non-inverting input; it is determined only by the input voltage source connected to E4.

&quot;What is the voltage gain when E1 is connected?&quot; means &quot;What is the voltage gain when E1 is connected to ground?&quot; So, the circuit is a non-inverting amplifier consisting of R1, RFB and the op-amp... and the OP's formula is correct.

&quot;What is the voltage gain when E2 is connected?&quot; means &quot;What is the voltage gain when E2 is connected to ground?&quot; So, again the circuit is a non-inverting amplifier now consisting of R2, RFB and the op-amp... and again the OP's formula is correct.

&quot;Neither E1 or E2 is connected.&quot; means &quot;Both E1 and E2 are not grounded&quot;. Then the circuit is simply a voltage follower where RFB does not play any significant role. So,

&quot;My guess is that there is no there would be no voltage gain, since there are no resistance on the top half of the circuit?&quot; means &quot;My guess is that there would be voltage gain of one, since there is no grounded resistance (neither R1 nor R2) on the input of the circuit?&quot;


The purpose of the professor's test is probably to determine whether

OP knows what the difference between the op-amp non-inverting amplifier and op-amp follower is;

OP understands what is ideal and what real voltage source... and what is the resultant voltage when they are connected in parallel.