Determine the volume of the room. For example if the room is 10ftx10ftx10ft the volume would be 1000ft cubed. The average mens basketball has a diameter of 25cm. There is approximately 30cm in a foot. Therefor you could fit one inflated basketball in a 1 foot cubed space. Therefor you could fit 1000 inflated basketballs inside a room with a volume of 1000 ft cubed.

If we can deflate the basketballs and flatten them down to one inch thick, this would allow us to place 12 flattened basketballs in a 1 foot cubed space. Therefore you could fit 12,000 basketballs in a room with a volume of 1000 ft cubed.

It would be more complicated depending on the shape of the room but the process for figuring out the solution would be the same.

Depends ... should we assume they're inflated?