Question

Find the number of parallelograms formed if $10$ parallel lines in a plane are intersected by a family of $12$ parallel lines.

Answer 1

Find the number of parallelograms:

A combination is a grouping of outcomes in which the order does not matter.

The number of combinations of $\mathrm{n}$ things chosen $\mathrm{r}$ at a time is found by using the following formula,

${}^{\mathbf{n}}\mathbf{C}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\left(n-r\right)\mathbf{!}}$

Here, $\mathrm{n}$ is number of items in set and $\mathrm{r}$ is the number of items selected from the set.

We know that a parallelograms is a four sided figure which has two pairs of parallel side.

Two straight lines are used to make a parallelogram. $2$ lines from the $10$ line set and $2$ lines from the set of $12$ parallel lines.

The $2$ parallel lines can be chosen from the set of $10$ parallel lines in ${}^{10}\mathrm{C}_2$ method.

In ${}^{12}\mathrm{C}_2$ methods, the $2$ parallel lines can be chosen from a set of $12$ parallel lines.

So,

Number of parallelograms formed $={}^{10}\mathrm{C}_2\times {}^{12}\mathrm{C}_2$

$=\frac{10!}{2!8!}\times \frac{12!}{2!10!}$

$=\frac{10\times 9}{2}\times \frac{12\times 11}{2}$

$=2970$

Hence, the number of parallelogram formed is $2970$.