$N$ balls and $M$ boxes, probability of last $ i$ boxes are empty

Question

I encountered this problem. There are $M$ boxes and $N$ balls. Balls are thrown to the boxes randomly with probability of $\frac1M$. The boxes are numbered $1, 2, 3, ..., M$. what is the probability of last $i$ slots are empty, $i = 1, 2, 3, ...,M-1$?

I appreciate any insight on the problem.

Answer 1

There are $M^N$ functions from the set of balls to the set of boxes, all equally likely. The number of functions that miss $i$ specific boxes is $(M-i)^N$.

Equivalently, the probability that the first ball misses $i$ specified boxes is $\frac{M-i}{M}$. By independence, the probability they all do is $\left(\frac{M-i}{M}\right)^N$.