Question

Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

Answer 1

Since,

Number of parallelograms when '$m$' number of parallel lines intersects with '$n$' number of parallel lines $={}^m{C}_2\times {}^n{C}_2$

There are two sets of parallel lines, one with $4$ parallel lines and one with $3$ parallel lines.

Required number of parallelograms

$={}^4{C}_2\times {}^3{C}_2$

$=\frac{4!}{2!(4-2)!}\times \frac{3!}{2!(3-2)!}$ [Since, ${}^n{C}_r=\frac{n!}{r!(n-r)!}$]

$=\frac{4!}{2!2!}\times \frac{3!}{2!1!}$

$=\frac{4\times 3\times 2\times 1}{2\times 1\times 2\times 1}\times \frac{3\times 2\times 1}{2\times 1\times 1}$

$=6\times 3$

$=18$

Required number of parallelograms $=18$