Find the number of all onto functions from the set $\{1, 2, 3, ...., n\}$ to itself.

The number of onto functions that can be defined from a finite set X containing n elements on to a finite set Y containing n elements.

Let X : $\{1, 2, ...., n\}$ and Y : $\{1, 2, 3, ...., n \}$

One of the elements of set X(say 1) has any one of the pre-image 1, 2, ..., n i.e. n ways.

In similar way, the element (say 2) in (n$-$1) ways

$\therefore$ Total number of possible ways$=$n (n$-$1 ) (n $-$2) …..3.2.1

$=$ n!