Rectangular tiles

In order for the center rectangle not to touch the edges of the $9 \times 7$ outer rectangle, it must stay within a $7 \times 5$ region. That means that the center rectangle can have $7 \times 5=35$ distinct sizes. An $a \times b$ center rectangle can occupy $(7-a+1)\times(5-b+1)$ positions inside the outer rectangle. See the table below:

The sum of all the numbers in this table is clearly: $\frac{7(7+1)}{2} \times \frac{5(5+1)}{2}=28 \times 15=420$

For each of the 420 possible center rectangle positions, there are four lines coming from the four corners of the rectangle to divide the border into four rectangles. Each of these lines can be either vertical or horizontal, so there are 16 unique dispositions of the four border rectangles for each center rectangle. Therefore, there are $420 \times 16=\boxed{6720}$ possibilities in total.