How many shapes

Choosing rectangles(including squares) are just like choosing $2$ vertical lines and $2$ horizontal lines.

In a $9\times 5 ~\text{rectangle}$ we have $6$ vertical lines and $10$ horizontal lines

So, $2$ vertical lines can chosen in ${10\choose 2}$ ways from $10$ vertical lines and $2$ horizontal lines can chosen in ${6\choose 2}$ ways from $6$ horizontal lines

So, in a $9\times 5 ~\text{rectangle}$ number of rectangles (including squares) possible are:

$\begin{aligned} {10\choose 2}\times {6\choose 2}&= \dfrac{10!}{2!\cdot 8!}\times \dfrac{6!}{2!\cdot 4!} \\& =\dfrac{10\times 9}{2}\times \dfrac{6 \times 5}{2}\\&=\boxed{675} \end{aligned}$