How many rectangles in a 7x6 rectangle? I'm working on it now by carefully listing all the possibilities.
Is there a fast way to do this? And what's the answer anyways?
 A: select two horizontal lines from the available $8$ horizonal lines $\dbinom{8}{2}$ ways
select two vertical lines from the available 7 vertical lines $\dbinom{7}{2}$ ways
Required count of rectangles
$\dbinom{8}{2} \times \dbinom{7}{2}$
A: This is the way I solved this problem before I read Kiran's answer that exposed a faster method:
The grid that encloses the $7 \times 6$ rectangles consists of $8 \times 7$ lines and $8 \cdot 7 = 56$ points.  Two of such points define the diagonal of a rectangle and therefore exactely one rectangle.  There are $56 \choose 2$ pairs of points. But if two points lie on the same horizontal line or on the same vertical line this is a degenerated rectangle that we won't count. On  a horizontal line we can choose $8 \choose 2$ pairs of points and there are $7$ horizontal lines. On  a vertical line we can choose $7 \choose 2$ pairs of points and there are $8$ vertical lines. We have to subtract them from our number. The remaining number counts each rectangle twice  because a rectangle has two diagonale. So there are
$$\left ({56 \choose 2} - 7 \cdot {8 \choose 2} - 8 \cdot {7 \choose 2}\right)/2$$
rectangles.
