How many spheres can fit inside this larger sphere? I would like to know if there is a way to do the following: calculate the maximal number of spheres of unit radius that can fit inside a sphere of radius 200 times the unit radius.
This is a generalisation of a question that was asked in a biology class. I was wondering if there exist some theorems on this, since I don't know how to start on it.

There is also a packing arrangement known as Random Close Pack. RCP depends on the object shape - for spheres it is 0.64 meaning that the packing efficiency is 64% (as you can also see in Jack D'Aurizio's link). Therefore, if the balls are randomly distributed, then you can fit approximately $0.64 \cdot \frac{\frac{4}{3}\pi (20r)^3}{\frac{4}{3}\pi r^3} \approx 5120 $, which is far from the highest estimation but still pretty good.
