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

$\begin{array}{1 1} n ! \\ n \\ 1 \\ 0 \end{array}$

Toolbox:
• A function f is onto in $f:A \to A$ if for each element $x \in A$ there exists $y \in a$ such that $f(x)=y$
• No of onto functions from a given set {1,2.....x} itself is the permutation of n symbols 1,2.......n
Total number of maps from {1,2,...n} to itself in permutation of n symbols which is 'n!'.

edited Mar 20, 2013