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Problem 654

AIME late

Combinatorics Difficulty 5.2 Find the answer

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.

Your answer

Official solution

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.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.