The number of five digit numbers that can be formed with $0,1,2,3,5$ which are divisible by $25$ is

The required $5$ digit numbers should have $25$ or $50$ in the end.
The total number of $5$-digit numbers containing $25$ in the end will be ...(no repetition of digits)
$3!$.
Of these the number of numbers containing $0$ from the beginning are $2!,2!$ due to internal permutations.
Hence, total number of $5$ digit number with $25$ in the end using the given set of $5$ number are
$3!-2!$ ...(no repetition of digits)
$=4$ ...(i)
Now the total number of $5$ digit numbers with $50$ at the end
$=3!$ ....(no repetition of digits)...(ii)
Hence, the total number of $5$ digit numbers formed using $0,1,2,3,5$ without repetitions of digits, and divisible by $25$ are
$3!+4$
$=6+4$
$=10$