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?
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?
A number is divisible by $10$ if its units digits is $0 .$
Therefore, $0$ is fixed at the units place.
Therefore, there will be as many ways as there are ways of filling $5$ vacant places $\boxed{}\,\boxed{}\,\boxed{}\,\boxed{}\,\boxed{}\,\boxed0\,$ in succession by the remaining $5$ digits (i.e., $1,3,5,7$ and $9$ ).
The $ 5$ vacant places can be filled in $5 !$ Ways.
Hence, required number of $6 -$ digit numbers $=5 !=120$
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