Number and probability of random allocations of n processes to n servers
The number $n^n$ can be easily obtained if you think the problem in the other direction: each process has $n$ choices and therefore $n$ processes has $n^n$ possible choices in total. Of course it is impossible that two servers have $n$ processes. If a process can be assigned to more than one server, the number of possible assignments is far more than $n^n$.