Question

Let $A$ be a set of $n$ distinct elements. Then the total number of distinct functions from $A$ to $A$ is ____ and out of these ____are onto functions.

Answer 1

Let $A=\{a_1,a_2,...a_n\}$.

We can associate each $a_i(1\le i\le n)$ with any of $a_1,a_2,...a_n$.
Thus, the number of function from $A$ to $A$ is $n^n$.

Since $A$ is finite, a function $f:A\to A$ is onto if and only if it is one-to-one.
$\therefore$ For $a_1$ we have $n$ choices; $a_2$, there are $(n-1)$ choices and so on.
Thus, required number of onto functions is $n(n-1)...2.1=n!$