Three identical coils C_ 1 , C_ 2 and C_ 3 are closely placed such...

Let the common axis be the z-axis.
$C_1$ is at $z = -d$, $C_2$ is at $z = 0$, and $C_3$ is at $z = +d$.
Current in $C_1$ is anti-clockwise (ACW), producing a magnetic field $B_1$ in the $+z$ direction at $C_2$.
Current in $C_3$ is clockwise (CW), producing a magnetic field $B_3$ in the $-z$ direction at $C_2$.
The net magnetic field at $C_2$ is $B_{net} = B_1 - B_3$. Since $C_1$ and $C_3$ are equidistant and carry equal current, initially $B_{net} = 0$.

To induce a clockwise current in $C_2$, Lenz's law states that the induced current must oppose the change in flux. A clockwise induced current produces a magnetic field in the $-z$ direction.
This happens if there is an increasing flux in the $+z$ direction or a decreasing flux in the $-z$ direction.

Case 1: $C_1$ moves towards $C_2$. The distance decreases, so $B_1$ (in $+z$ direction) increases. This causes an increasing flux in the $+z$ direction.

Case 2: $C_3$ moves away from $C_2$. The distance increases, so $B_3$ (in $-z$ direction) decreases. This causes a decreasing flux in the $-z$ direction.

Both actions (moving $C_1$ towards $C_2$ and moving $C_3$ away from $C_2$) result in a net change in flux that induces a clockwise current in $C_2$ to oppose the change.
Therefore, option (3) is correct.