How many five-digit numbers, which do not contain identical digits, can be written by means of the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ and $9$?
Given digits: $\;$ $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ and $9$ $\;$ i.e. $\;$ $9$ digits
Since the digits are not repeated,
Ten thousand's place can be selected in $9$ ways
Thousand's place can be selected in $8$ ways
Hundred's place can be selected in $7$ ways
Ten's place can be selected in $6$ ways
Unit's place can be selected in $5$ ways
$\therefore \;$ Number of five-digit numbers that can be formed with the given digits so that the digits are not repeated
$= 9 \times 8 \times 7 \times 6 \times 5 = 15120$ numbers