We pave an equilateral triangle with side $n$ using $n^{2}$ small equilateral triangles with side 1. How many parallelograms does the resulting figure contain?
First, parallelograms have only 3 possible orientations, and by symmetry there are as many parallelograms of each orientation. We content ourselves with counting those with a tip pointing upwards. For this, we extend our tiling by one line as shown in the drawing. For each of the parallelograms, we extend its sides, and we get four intersections with the line added below our triangle. We verify that this gives a bijection between the parallelograms pointing upwards and the sets of four points on our grid on the last line added. Since there are \(n+2\) grid points on this line, there are \(\binom{n+2}{4}\) parallelograms pointing upwards, and thus \(3 \binom{n+2}{4}\) parallelograms in total.

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