A set of 6 parallel lines is cut by another set of 5 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 1
The correct answer is Option (1) → 150
A parallelogram has two pairs of parallel sides, so it is fixed by choosing 2 lines from each set; the two choices are multiplied.
2 lines from the set of 6: \(\displaystyle ^6C_2=\frac{6\times5}{2}=15\) ways
2 lines from the set of 5: \(\displaystyle ^5C_2=\frac{5\times4}{2}=10\) ways
Parallelograms \(=15\times10=150\)
Why the other options are incorrect
Option (2) 30 — Incorrect. This is \(6\times5\), the number of points where the lines cross, not parallelograms.
Option (3) 300 — Incorrect. This is \(2\times150\), which counts each parallelogram twice.
Option (4) 55 — Incorrect. This is \(^{11}C_2\), the number of ways to pick any 2 of the 11 lines; two lines alone do not make a parallelogram.