Students A and B connect the two resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\]given to them in the manners as given below and then insert them at X and Y into the measuring circuit shown below.

We can say that –
A) Both the students will determine the equivalent resistance of the series combination of \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\].
B) Both the students will determine the equivalent resistance of the parallel combination of \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\].
C) Student A will determine the equivalent resistance of the series combination while the student B will determine the equivalent resistance of the parallel combination of \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\].
D) Student A will determine the equivalent resistance of the parallel combination while the student B will determine the equivalent resistance of the series combination of \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\].

Hint: We have to determine the type of combination set up the students with the resistors they are offered with. The arrangement from the figures A and B will give the type of network they have used to get the equivalent resistance in the measuring circuit.

Complete step by step solution:
We can analyse the circuits arranged by the two students with the resistors. The connection between the points will help us understand the type of network they did arrange.

We can see from the figure that the resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\]are connected to each other only once. This implies that the resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\] are connected in series with each by the student A.

The resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\]are connected to each other at two points in the arrangement done by the student B. The resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\]are connected to each other from points $a_{1}$ to $d_{1}$ and from $b_{1}$ to $c_{1}$, which is in turn connected to X and Y respectively. Therefore, we can conclude that the resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\]are connected parallel to each other by the student B.
Now, when the students employ their respective connections in the measuring circuit, we can say that the –
The student A will get the equivalent resistance of the series combination of the resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\].
The student B will get the equivalent resistance of the parallel combination of the resistors \[{{\text{R}}_{\text{1}}}\text{ and }{{\text{R}}_{\text{2}}}\].

So, the correct answer is “Option C”.

Note: When two resistors are connected, they can either be connected in series combination or parallel combination only. The number of nodes connecting them together determine the type of combination they are designed in, two for parallel and one for series.