Solution:

Given, a ball with charge -50e is placed at the centre of hollow spherical shell has a net charge of -50e.

Charge on the shell's outer surface can be determined by using Gauss's law.

1. Gauss's Law:

Gauss's law states that the electric flux through any closed surface is directly proportional to the electric charge enclosed within it.

Φ = Q/ε₀

Where,
Φ - Electric flux
Q - Charge enclosed within the surface
ε₀ - Permittivity of free space

2. Applying Gauss's Law:

As per the given problem statement, a ball with charge -50e is placed at the centre of the hollow spherical shell, which has a net charge of -50e.

Since the electric field inside the shell is zero, it can be considered as a closed surface for the purpose of applying Gauss's law.

The charge enclosed within the surface is -50e (charge on the ball).

Therefore, the electric flux through the surface is also -50e/ε₀.

3. Determining the Charge on the Shell's Outer Surface:

As per Gauss's law, the electric flux through any closed surface is directly proportional to the electric charge enclosed within it.

Since the electric flux through the surface is -50e/ε₀, the charge on the shell's outer surface must also be -50e.

Therefore, the correct option is (C) -50e.

Conclusion:

Hence, the charge on the shell's outer surface is -50e, as determined using Gauss's law.