What is the number of odd integers between 1000 and 9999 with no digit repeated?

Case I When unit digit can be $1,3,5$ or $7 \&$ digit at thousand's place can be $1,2,3,4,5,6,7$ or 8 . No. of ways digits can be filled are: \begin{array}{|l|l|l|l|}\hline 7 & 8 & 7 & 4 \\\hline\end{array} Total no's $=7 \times 8 \times 7 \times 4=1568$. Case II When unit digit can be $9 \&$ digit at thousand's place can be $1,2,3,4,5,6,7$ or 8 . No. of ways digits can be filled are: \begin{array}{|l|l|l|l|}\hline 8 & 8 & 7 & 1 \\\hline\end{array} Total no's $=8 \times 8 \times 7 \times 1=448$. Case III When unit digit can be $1,3,5$ or $7 \&$ digit at thousand's place can be 9 . No. of ways digits can be filled are: \begin{array}{|l|l|l|l|}\hline 1 & 8 & 7 & 4 \\\hline\end{array} Total no's $=1 \times 8 \times 7 \times 4=224$. $\begin{aligned} \therefore \quad \text { Number of odd digits between } 1000 \& 9999 & \text { with no } \\ \text { digit repeated } & =1568+448+224 \\ & =2240 .\end{aligned}$