Question

The number of all onto functions from the set $A=\{1,2,3,\dots n\}$ to itself is

• A. $n!$
• B. $2n!$
• C. $n{!}^2$
• D. $n{!}^n$

Answer 1

The correct option is A $n!$

$A=\{1,2,3,\cdots n\}$
Element Number of possible pairing
$1$ $n$
$2$ $n-1$
$3$ $n-2$
$\cdot$ $\cdot$
$\cdot$ $\cdot$
$n-1$ $2$
$n$ $1$
Total number of onto functions
$=n\times n-1\times n-2\times \cdots \times 2\times 1=n!$