What is the number of odd integers between 1000 and 9999 with no...

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|}\hline7 & 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|}\hline8 & 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|}\hline1 & 8 & 7 & 4 \hline\end{array}Total no's $=1 \times 8 \times 7 \times 4=224$. $$\begin{aligned} \therefore \quad \text{ Number of odd digits between } 1000 & 9999 & \text{ with no } \\ \text{ digit repeated } & =1568+448+224 \\ & =2240 .\end{aligned}$$