Step-by-step Solution:

1. Analyze the condition for a helical path: A charged particle follows a helical path when it moves in a uniform magnetic field, and its velocity vector has components both parallel ($\overrightarrow v_\parallel$) and perpendicular ($\overrightarrow v_\perp$) to the magnetic field ($\overrightarrow B$). The axis of the helix is always parallel to the direction of the magnetic field vector $\overrightarrow B$.

2. Identify the required direction of the magnetic field: The question states that the axis of the helical path must be along the positive z-direction. This implies that the magnetic field $\overrightarrow B$ must be directed along the z-axis. Looking at the options in Column 3:

   • (P) $\overrightarrow B = - {B_0}\widehat x$ (along negative x-axis)
   • (Q) $\overrightarrow B = {B_0}\widehat x$ (along positive x-axis)
   • (R) $\overrightarrow B = {B_0}\widehat y$ (along positive y-axis)
   • (S) $\overrightarrow B = {B_0}\widehat z$ (along positive z-axis) Only option (S) has the magnetic field along the z-axis. Therefore, the correct combination must include (S) from Column 3.

3. Evaluate the given options: Now, let's check which of the multiple-choice options includes (S) from Column 3.

   • A: $\left( {{\rm I}V} \right)\left( i \right)\left( S \right)$ - This option includes (S).
   • B: $\left( {{\rm I}{\rm I}} \right)\left( {ii} \right)\left( R \right)$ - This option includes (R), so the axis would be along the y-direction. Incorrect.
   • C: $\left( {{\rm I}{\rm I}{\rm I}} \right)\left( {iii} \right)\left( P \right)$ - This option includes (P), so the axis would be along the x-direction. Incorrect.
   • D: $\left( {{\rm I}V} \right)\left( {ii} \right)\left( R \right)$ - This option includes (R), so the axis would be along the y-direction. Incorrect.

4. Conclusion from elimination: Based on the direction of the magnetic field, only option A is plausible. Let's verify that this combination indeed results in the described motion.

5. Detailed analysis of Option A: (IV)(i)(S):

   • Particle and Initial Velocity (IV): Proton with charge $q = +e$ and initial velocity $\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$. The initial velocity is purely in the xy-plane (perpendicular to the z-axis).
   • Electric Field (i): $\overrightarrow E = {E_0}\widehat z$. The electric field is along the positive z-axis.
   • Magnetic Field (S): $\overrightarrow B = {B_0}\widehat z$. The magnetic field is along the positive z-axis.

6. Calculate the Lorentz Force: The total force on the proton is given by the Lorentz force law: $\overrightarrow F = q(\overrightarrow E + \overrightarrow v \times \overrightarrow B)$

7. Analyze the components of motion:

   • Motion parallel to B (z-direction): The electric field $\overrightarrow E$ is parallel to the magnetic field $\overrightarrow B$. The electric force on the proton is $\overrightarrow F_E = q\overrightarrow E = eE_0\widehat z$. This force is constant and directed along the positive z-axis. The magnetic force, $\overrightarrow F_B = q(\overrightarrow v \times \overrightarrow B)$, is always perpendicular to $\overrightarrow B$, so it has no z-component. Therefore, the net force in the z-direction is $F_z = eE_0$. This causes a constant acceleration $a_z = eE_0/m$ in the positive z-direction. Since the proton starts at the origin with $v_z(0)=0$, it will move along the positive z-axis.

   • Motion perpendicular to B (xy-plane): The initial velocity of the proton, $\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$, is entirely perpendicular to $\overrightarrow B$. The magnetic force component in the xy-plane, $q(\overrightarrow v_\perp \times \overrightarrow B)$, acts as a centripetal force, causing the proton to execute circular motion in the xy-plane. The electric field has no component in this plane.

8. Combine the motions: The resulting trajectory is a superposition of circular motion in the xy-plane and accelerated linear motion along the positive z-axis. This combined motion describes a helical path whose axis is the z-axis. Since the particle accelerates in the +z direction, it moves along the positive z-axis. The pitch of the helix increases with time, but it is still a helical path.

9. Final Conclusion: The combination (IV)(i)(S) correctly describes a particle moving in a helical path with its axis along the positive z-direction. Therefore, option A is the correct answer.