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A set of 10 parallel lines is intersected by another set of $m$ parallel lines. If the total number of parallelograms formed by these two sets of lines is 675 . Then $m$ is equal to
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$6$
Given, 10 parallel lines and $m$ parallel lines since one parallelogram can be formed by the intersection on one pair of parallel lines with another pair of another set of parallel lines.
$\therefore$ Number of parallelograms $={ }^{10} C_2 \cdot{ }^m C_2$
$\Rightarrow \quad 675=\frac{10 \times 9}{2} \times \frac{m(m-1)}{2}$
$\Rightarrow \quad 30=m(m-1) \Rightarrow m^2-m-30=0$
$\Rightarrow \quad(m-6)(m+5)=0 \Rightarrow m=6$
$\therefore$ Number of parallelograms $={ }^{10} C_2 \cdot{ }^m C_2$
$\Rightarrow \quad 675=\frac{10 \times 9}{2} \times \frac{m(m-1)}{2}$
$\Rightarrow \quad 30=m(m-1) \Rightarrow m^2-m-30=0$
$\Rightarrow \quad(m-6)(m+5)=0 \Rightarrow m=6$
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