Mutual inductance is a measure of the magnetic coupling between two coils. It determines the extent to which a change in current in one coil induces a voltage in the other coil. The mutual inductance between two magnetically coupled coils depends on various factors. Let's discuss each of these factors in detail:

a) Number of turns:
- The mutual inductance between two coils is directly proportional to the number of turns in both coils.
- Increasing the number of turns in either coil will result in an increase in mutual inductance.
- This is because more turns create a stronger magnetic field, leading to a higher coupling between the coils.

b) Permeability of the core:
- The mutual inductance also depends on the permeability of the core material used in the coils.
- Permeability is a measure of how easily a material can be magnetized.
- Higher permeability allows for better magnetic coupling between the coils, resulting in increased mutual inductance.
- Therefore, the choice of core material with a high permeability can enhance the mutual inductance.

c) Cross-sectional area of their common core:
- The mutual inductance is also influenced by the cross-sectional area of the common core shared by the two coils.
- Increasing the cross-sectional area of the core increases the magnetic flux passing through the coils, leading to a higher mutual inductance.
- This is because a larger area allows for a greater number of magnetic field lines, resulting in better magnetic coupling.

d) All of the above:
- Considering all the factors mentioned above, it is clear that the mutual inductance between two magnetically coupled coils depends on the number of turns, permeability of the core, and cross-sectional area of their common core.
- All these factors contribute to the magnetic coupling between the coils, affecting the mutual inductance.

In conclusion, the mutual inductance between two magnetically coupled coils depends on the number of turns in the coils, the permeability of the core material, and the cross-sectional area of their common core. These factors determine the extent of magnetic coupling and influence the induction of voltage in one coil due to a change in current in the other coil.