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. 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

Sol. Explanation:

(1) is wrong since the correct answer must become 0 for

n = 3.

(2) Is also wrong since the correct answer must become 3 for n = 4

(3) Seems to be correct.

Alternative Solution : n lines will give ⁿC₂ points of intersections. These points can be again joined to get $nC2C_{2}$line segments which will contain n. ^n–1C₂ old lines.

̃ The number of fresh lines = $nC2C_{2}$– n. ^n–1C₂

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