How many parallelograms can be formed when a parallelogram is cut by $2$ sets of $n$ parallel lines?

Question

A parallelogram is cut by two sets of n parallel lines parallel to the sides of the parallelogram. The number of parallelogram thus formed is..??

I think we can do it by combinatorics.. But I'm not quite sure... Help me out please.

Answer 1

Consider parallelogram $ABCD$:

A parallelogram is determined by choosing two pairs of opposite sides. If you introduce $n$ lines parallel to the sides of the parallelogram, you have $n + 2$ sides from which to choose the two sides parallel to $\overline{AB}$ and $\overline{CD}$ (including $\overline{AB}$ and $\overline{CD}$ themselves) and also have $n + 2$ sides from which to choose the two sides parallel to $\overline{AD}$ and $\overline{BC}$ (including $\overline{AD}$ and $\overline{BC}$ themselves). Thus, there are $$\binom{n + 2}{2}\binom{n + 2}{2} = \binom{n + 2}{2}^2$$ parallelograms that can be formed from the grid that results when each side of the parallelogram is cut by $n$ lines.

Answer 2

Hint: You can squash the parallelogram to make a square, which may as well have the sides vertical and horizonal. Now cut the square with $n$ vertical lines. How many pieces? Cut them all with $n$ horizontal lines.... Draw a picture.