Probability problem involving mappings of a finite set into itself. One mapping is selected at random from all the mappings of the set $\{1,2,\ldots,n\}$ into itself. What is the probability that 
$(i)$ a specified element $i$ is transformed into another specified element $j$?
$(ii)$ the elements $i_1,i_2,\ldots,i_h$ are transformed into the elements $j_1,j_2,\ldots,j_h$ respectively?

For (ii) suppose that the $i_k$ are all distinct, and the $j_k$ are all distinct.
There are $n(n-1)(n-2)\cdots(n-h+1)=n!/(n-h)!$ $h$-tuples of distinct elements
from $\{1,2,\ldots,n\}$. It is equally likely that your first $h$-tuple is taken
to any one of them, so the probability you seek is $(n-h)!/n!$.
