Question

There are $10$ straight lines in a plane no two of which are parallel and no three are concurrent. The points of intersection are joined, then the number of fresh lines formed are

• A. 630
• B. 615
• C. 730
• D. 600

Answer 1

Answer: (A)

630

There are ${{}^{10}C}_2$ ways of choosing one point of intersection (pick two distinct lines from the set of 10);
There are ${}^{(10-2)}{C}_2$ ways of picking the second point of intersection to determine the new line (pick two distinct lines from among the remaining $(10-2)$).
The two points uniquely determine a new line. But you've counted each line twice, since the order in which you select the two points does not matter. So the total number should be
$\frac{1}{2}\times {{}^{10}C}_2\times {{}^{8}C}_2=630$