21.1.3 ${ }^{\star \star}$ Divide each side of the equilateral $\triangle A B C$ into $n$ equal parts, and draw lines parallel to the sides through the division points. Find the number of parallelograms in the resulting figure.
Consider the number of parallelograms without any side parallel to $BC$. Each such parallelogram corresponds one-to-one with four different points on $B^{\prime} C^{\prime}$ (as shown in the figure), so there are $\mathrm{C}_{n+2}^{4}$ parallelograms. Similarly, the number of parallelograms without any side parallel to $AC$ or $AB$ is also $\mathrm{C}_{n+2}^{4}$ each. Therefore, there are a total of $3 \cdot \mathrm{C}_{n+2}^{4}$ parallelograms.