Give two definitions of mutual inductance, give its units and write factors on which its value depends.

(N/A) The magnetic flux $\Phi_{2}$ linked with coil-$2$ when current $I_{1}$ flows through coil-$1$ is given by $\Phi_{2} = M_{21} I_{1}$.
Taking $I_{1} = 1 \text{ unit}$, we get $\Phi_{2} = M_{21}$. Thus, "The magnetic flux linked with one of the coils per unit current passing through the other coil is called the mutual inductance of the system."
The induced emf $\varepsilon_{2}$ in coil-$2$ is given by $\varepsilon_{2} = -M_{21} \frac{dI_{1}}{dt}$.
When $\frac{dI_{1}}{dt} = 1 \text{ unit}$, we get $\varepsilon_{2} = M_{21}$. Thus, "The mutual emf generated in one of the two coils due to a unit rate of change of current in the other coil is called the mutual inductance of the system."
The $SI$ unit of mutual inductance is the henry $(H)$, where $1 \text{ H} = 1 \text{ Wb A}^{-1} = 1 \text{ V s A}^{-1}$.
The value of mutual inductance depends on:
$(1)$ Shape and size of the coils.
$(2)$ Number of turns in the coils.
$(3)$ Distance between the coils.
$(4)$ Relative orientation (angle of inclination) of the coils.
$(5)$ Magnetic permeability of the core material on which the coils are wound.