A box (plural: boxes) is a container with rigid sides used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides (typically rectangular prisms). Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture) and can be used for a variety of purposes, from functional to decorative.
Boxes may be made of a variety of materials, both durable (such as wood and metal) and non-durable (such as corrugated fiberboard and paperboard). Corrugated metal boxes are commonly used as shipping containers.
Boxes may be closed and shut with flaps, doors, or a separate lid. They can be secured shut with adhesives, tapes, string, or more decorative or elaborately functional mechanisms, such as catches, clasps or locks.
I made this question up so I have no guarantee that there is a clean answer. It seems like there should be a simple approach though, I’m just not seeing it.
First attempt:
Find the chance that only the first k boxes are left empty. Then we can multiply by ##{n \choose k}## to get the total...