Question

How many $6$-digit numbers can be formed from the digits $0,1,3,5,7$ and $9$ which are divisible by $10$ and no digit is repeated$?$

Answer 1

Finding $6$ -digit numbers divisible by $10$ formed by using the digits $0,1,3,5,7$ and $9$ when no digit is repeated.

a number is divisible by $10$ then $0$ is at the unit place.
_ _ _ _ _ 0
So, remaining digits are $1,3,5,7$ and $9$

$n=5$ and $r=5$

$\therefore$ Number of 6-digit number divisible by $10={}^5{P}_5$

$=\frac{5!}{(5-5)!}$

$=\frac{5\times 4\times 3\times 2\times 1}{0!}=120$

Hence, required $6$-digit numbers are $120$.