We consider two activities each performed with the following objects :
balls all distinct in sizes.
black boxes (all distinct)
white boxes (all distinct)
Define activity as follows :
Choose any balls from the collection of distinct balls. We call these balls chosen and the remaining balls unchosen.
Distribute the balls in the boxes in such a way that : Every ball goes to a box.
None of the unchosen balls goes to a black box.
None of the chosen balls goes to a white box.

Let be the set of all possible outcomes of the activity .
Let be the total number of ways to complete the activity .

Define activity as follows :
Distribute the balls in such a way that every ball goes to some box.

Let be the set of all possible outcomes of the activity .
Let be the total number of ways to complete the activity .

Then
and
and
So,
and thus, and thus

This completes the solution.