A set of 7 parallel lines is cut by another set of 4 parallel lines (the lines of the second set are not parallel to the lines of the first set). How many parallelograms are formed by these lines?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 126
A parallelogram needs two lines from each set: a pair of parallel sides from the first set and a pair from the second. The two choices are multiplied.
Two lines from the 7: \(\displaystyle ^7C_2=\frac{7\times6}{2}=21\)
Two lines from the 4: \(\displaystyle ^4C_2=\frac{4\times3}{2}=6\)
Parallelograms \(=21\times6=126\)
Why the other options are incorrect
Option (1) 28 — Incorrect. This is \(7\times4\), the number of crossing points, not parallelograms.
Option (2) 330 — Incorrect. This is \(^{11}C_4\), any 4 of the 11 lines; 4 lines from the same set make no parallelogram.
Option (3) 252 — Incorrect. This is twice the count; each choice of 2 + 2 lines gives exactly one parallelogram.