How many odd numbers of $5$ digits can be formed with the digits $0,2,3,4,5$ without repetition of any digit?
I noticed that the last number can be filled in $2$ ways and first place can be filled in $3$ ways. Consequently, 2nd 3rd and 4th places can be filled in $3$, $2$ and $1$ ways respectively. Is there any flaw in this reasoning? Please guide further.