Probabilty + Permutations + Functions

There is a set $A = [ 1,2,3,..... n]$ . Then all mapping from $A$ to $A$ can be given by $n^n$ (Total). Becasue for $1$ there are $n$ numbers which can be it's image, same for all $n$ digits so all are multiplied hence $n^n$ . Now we know that for $Bijective$ it is one one + onto . That means that $1$ will get any value in the $set A = [1,2,3.....n]$ . Or $n$ values, Now $2$ will get $(n-1)$ values as $1$ is used by $1$ .Hence total bijective mapping is $n!$ . Now probability is $n!/n^n$