1.92 In the figure, the side length of the large triangle is 3, and the number of parallelograms formed by the lines passing through each grid point in the figure is \( f(3) = 15 \). In the general case, find the number of parallelograms \( f(n) \) in a triangle with side length \( n \).
[Solution] Let the number of parallelograms whose sides are parallel to the sides $AB, BC$ of $\triangle ABC$ (with side length $n$) be $g(n)$.

On the extensions of $BA, BC$, take points $E, F$ respectively such that $BE = BF = n+1$, then $EF = n+1$, and each side of the above parallelograms intersects $EF$ at exactly one point. These 4 points are distinct and are all lattice points on $EF$ (i.e., the points that divide $EF$ into $n+1$ equal segments).

Conversely, from the $n+2$ lattice points on $EF$ (including the endpoints $E, F$), choose any four points. Draw lines parallel to $AB$ through the two points closer to $E$, and draw lines parallel to $BC$ through the other two points, and a parallelogram as described above is obtained. Therefore,
$$
g(n) = C_{n+2}^{4},
$$

Thus,
$$
f(n) = 3 \times g(n) = 3 C_{n+2}^{4}.
$$