Calculate empty labeled boxes in labeled boxes-balls problem

Question

The problem is to count the number of times that K boxes are empty when n labeled balls are distributed into m labeled boxes. So, if we have n balls and m boxes, we have $m^{n}$ possibilities. I would like to count the number of times that box 1 and/or box 2 for example are empty.

For example, how many times box 1 or box 4 (or both) are empty when we distribute 10 balls between 10 boxes?. There are $10^{10}$ possibilities. How can it be calculated?

Answer 1

Hint: To ensure that some boxes are empty the balls have to be distributed between the remaining boxes.

For example:

The number of ways to distribute 10 balls so that the box 1 is empty is $9^{10}$.

The number of ways to distribute 10 balls so that the boxes 1 and 4 are empty is $8^{10}$.

Keep in mind that the above numbers count also the events with some other boxes being empty.