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?

Since there are no restrictions on these maps $f:\>[n]\to[n]$ there are $n^n$ of them.
(i) Given $i$, $j\in[n]$ the probability that $f(i)=j$ is $\>{1\over n}$.
(ii) Given a subset $A=\{i_1,i_2,\ldots, i_h\}\subset [n]$ with $|A|=h$ and values $j_k\in [n]$  $(1\leq k\leq h)$ the probability that $f(i_k)=j_k$ for $1\leq k\leq h$ is $\ n^{-h}\>$.
