# Difficult Geometry & Mensuration Solved QuestionAptitude Discussion

 Q. $S_1$and $S_2$ are two  sets of parallel lines.  The number of lines in $S_1$ is greater than the number of $S_2$ They intersect at 12 points.  The number of parallelograms that $S_1$ and $S_2$  may form is :
 ✖ A. 12 or 6 ✖ B. 8 or 4 ✖ C. 18 ✔ D. 18 or 15

Solution:
Option(D) is correct

Let there be $m$ lines is $S_1$ and $n$ lines in $S_2$.The lines will intersect in $mn$ points.

Therefore $mn = 12$.

Case (1), $m = 6, n = 2$ (or) $m = 3, n = 4,$

The number of parallelograms that can be formed is

$=^6C_2\times ^2C_2$

$=15\times 1$

$=15$
Case (2), $m = 4, n = 3,$

The number of parallelograms that can be formed is

$=^4C_2\times ^3C_2$

$=6\times 3$

$=18$