I need help with an equation?

There's a small technicality here: the definition of "transversal" does not guarantee that all transversals are parallel to each other.

But assuming that we are dealing with two sets of parallel lines, each set transversing the lines in the other, then we can assign variables

p = the number of parallel lines in one set; and

t = the number of parallel lines in the set transversing the first set.

I'm also unclear on whether we want just the set of smallest parallelograms, or whether we are counting those which enclose smaller parallelograms within them. So let's work it both ways.

If we're just counting the smallest ones, then there's one set of parallelograms between any pair of consecutive parallel lines in either set. There are one fewer such consecutive pairs in each set than the number of lines in the set.

So in this case, the number of parallelograms would be

(p-1) * (t-1)

where we have to specify that p>0 and t>0 to avoid a trivial error in the calculation if one of the sets has no lines at all.

I suspect, however, that what we really want is the total number of parallelograms, including those which contain sets of smaller parallelograms within them. In that case, every pair of lines in each set, whether consecutive or not, provides opposite sides for a set of parallelograms.

The number of pairs of lines in the set is the number of possible combinations of lines from the set, taken two at a time. If there are n lines in the set, this number is denoted nC2 or C(n,2) [whichever you prefer] and has the value

n(n-1)/2

because we can choose any of n lines for the first line of a pair, and any of the other n-1 lines for the second line, but if we exhaust all such possibilities we will have chosen each pair twice, with each line of the pair chosen first once and second the other time.

Again, we multiply the numbers of sides contributed by each set of lines, so our formula becomes

[p(p-1)/2] * [t(t-1)/2]

= p(p-1)t(t-1)/4

Notice in this case that we don't have to include the artificial restrictions p>0 and t>0, because if either p or t is 1 or 0 the result of the calculation is zero, which is correct.