It doesn't say "through the sphere." It says "through the center of the sphere." For example, drill a thin radial hole 6" inches deep through the center of a sphere of radius 5"; you will pass through the center and stop before you reach the far side. The volume remaining then depends on the radius of the drill bit, and the size of the sphere.

It also doesn't even state that the hole must enter the sphere. A large solid sphere that internally contains a six inch hole through its center would qualify too.

This was my objection when I first encountered the problem. Everyone else seemed to understand that the hole must pass through both sides of the sphere, but that's not stated or even implied in the problem.