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

A is set of n different elements.

$A$={$x_1,x_2,x_3,.............x_n$}

For one element there are n functions possible.

So for n elements ,

No. of functions possible$=n\cdot n\cdot n\cdot n\dots \dots \dots ..n$ [n times]

∴ number of functions from A to A $=n^n$

Onto functions: when every element of the image set has an inverse image in the first set.

If ,for $x_1$ there are n functions possible.

For $x_2$ there are (n-1) functions possible

Similarly for $x_3,x_4,.......x_n$

Number of Onto functions are $=n\times (n-1)\times (n-2).............1=n!$

∴ number of Onto functions from $A$ to $A=n!$