Question

There are n straight lines in a plane, no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is

• A. $\frac{n(n - 1)(n - 2)}{8}$
• B. $\frac{n(n - 1)(n - 2)(n - 3)}{6}$
• C. $\frac{n(n - 1)(n - 2)(n - 3)}{8}$
• D. None of these

Answer 1

Answer: (C)

$\frac{n(n - 1)(n - 2)(n - 3)}{8}$

Answer 2

Since no two lines are parallel and no three are concurrent, therefore n straight lines intersect at ⁿC₂ = N (say) points. Since two points are required to determine a straight line, therefore the total number of lines obtained by joining N points ^NC₂. But in this each old line has been counted ^n-1C₂ times, since on each old line there will be n – 1 points of intersection made by the remaining (n – 1) lines.

Hence the required number of fresh lines is

^Nc₂ – n. ^n-1C₂ = $\frac{N(N - 1)}{2} - \frac{n(n - 1)(n - 2)}{2}$

= $\frac{nC_{2}\left( nC_{2} - 1 \right)}{2} - \frac{n(n - 1)(n - 2)}{2} = \frac{n(n - 1)(n - 2)(n - 3)}{8}$.