Hint: Any rectangle is uniquely specified by the two horizontal and two vertical grid lines that enclose it.

For a two by three checkerboard, I drew 18 different rectangles which seem to correspond to

\(\displaystyle \left({4\atop 2}\right)\) * \(\displaystyle \left({3\atop 2}\right)\)

So I think the answer to the original problem would be

\(\displaystyle \left({8\atop 2}\right)\) * \(\displaystyle \left({5\atop 2}\right)\)

Is this correct reasoning?