How many parallelograms are formed when a set of 5 parallel lines intersect a set of 4 parallel lines?

This one is much more straightforward. There are 5C2 = 10 ways to choose two parallel lines from the set of five. There are 4C2 = 6 ways to choose two parallelograms from a set of four. Any parallelogram is uniquely determined by one pair of lines from the five, and one pair of lines from the four. Thus, the number of possible parallelograms is

(5C2)*(4C2) = (10)*(6) = 60