A recurrence for a combinatorial problem

Question

$N$ balls are tossed into $n$ boxes independently. Each ball has a $1/n$ chance of falling into any box.$$P_{N,n}(k):= Pr\{exactly\:k\:empty\:boxes\:after\:N\:balls\:thrown\:into\:n\:boxes\}$$ Show that:$$P_{N,n}(k) = (1-\frac{1}{n})^NP_{N,n-1}(k-1) + \sum_{i=1}^{N}\binom{N}{i}(\frac{1}{n^i})(1-\frac{1}{n})^{N-i}P_{N-i,n-1}(k)$$

Answer 1

Hint: The first term describes the situation in which box $n$ is empty. Term $i$ in the second sum (that should really go from $1$ to $N$) describes the situation in which box $n$ has $i$ balls.