The problem asks to calculate the moment of inertia (M.I.) of a system consisting of two discs of mass $M$ and radius $R$, connected by a weightless rod of length $2R$, about the perpendicular axis through its center.

Steps to Solve:

1. Moment of Inertia of each disc about its own center: The moment of inertia of a disc about an axis passing through its center and perpendicular to the plane is: $I_{\text{disc}} = \frac{1}{2} M R^2$

2. Using the Parallel Axis Theorem: The total moment of inertia is not just the moment of inertia of the discs about their own center but also takes into account their distance from the central axis. The parallel axis theorem states: $I_{\text{total}} = I_{\text{center}} + M d^2$ where $d$ is the distance of the disc's center from the axis of rotation.

3. Distance from the axis: The center of each disc is located at a distance $R$ from the center of the rod. So, the distance $d$ is $R$.

4. Moment of Inertia of each disc about the center of the rod: For each disc, using the parallel axis theorem: $I_{\text{disc (about the center)}} = \frac{1}{2} M R^2 + M R^2 = \frac{3}{2} M R^2$

5. Total Moment of Inertia: Since there are two discs: $I_{\text{total}} = 2 \times \left( \frac{3}{2} M R^2 \right) = 3 M R^2$

Conclusion:

The total moment of inertia of the system is $3 M R^2$. Thus, the correct answer is option (4).

Would you like further clarification or additional details? Here are some related questions you might find helpful:

1. What is the parallel axis theorem, and how is it derived?
2. How do we calculate the moment of inertia for different shapes?
3. Why do we add $M d^2$ when using the parallel axis theorem?
4. How does the moment of inertia change if the mass distribution is non-uniform?
5. How would the answer change if the rod had mass?

Tip: Always check whether the axis of rotation is through the center of mass or not, as this determines whether the parallel axis theorem is needed.