How many $5$-digit odd numbers can be formed using digits$0,1,2,3,4, 5$without repetition?

Making a $5$ -digit number is equivalent to fill $5$ places

To make odd numbers, fifth place can be filled by either of $1,3,5,$ i.e., $3$ ways.

Number of ways first place can be filled in =$4$ (excluding $0$ and the odd number used for the fifth place).

Similarly second, third and fourth places can be filled in $4,3,2$ ways respectively. Using fundamental principle of counting, total number of ways to fill $5$ places. = Total $5$ -digit odd numbers that can be formed =$4\times 4\times 3\times 2\times 3=288$ways..