Question

In a plane, a set of $8$ parallel lines intersect a set of ${}^{\prime }{n}^{\prime }$ parallel lines, that goes in another direction, forming a total $1260$ parallelograms. The value of ${}^{\prime }{n}^{\prime }$ is:

• A. $6$
• B. $10$
• C. $8$
• D. $12$

Answer 1

The correct option is B $10$

If m parallel lines in a plane are intersected by n parallel lines, the number of parallelograms that are formed $={m_C}_2\times {n_C}_2$

Given:-

No. of parallelograms $=1260$

$m=8$

$n=?$

$\therefore {8_C}_2\times {n_C}_2=1260$

$\Rightarrow 28\times \frac{n(n-1)}{2}=1260\{\because {n_C}_r=\frac{n!}{r!\times (n-r)!}\}$

$\Rightarrow n^2-n=90$

$\Rightarrow n^2-n-90=0$

$\Rightarrow n^2-10n+9n-90=0$

$\Rightarrow (n-10)(n+9)=0$

$n=10orn=-9$

$\because n\ne -ve$

$\therefore n=10$

Hence, the correct answer is 10.