Question

There are n straight lines in a plane, no 2 of which are parallel & no 3 pass through the same point. Their points of intersection are joined. Then the number of fresh lines introduced be $\frac{n(n-m)(n-m-1)(n-m-2)}{k}$.Find $m+k$ ?

Answer 1

Step 1st : Select 2 lines out of n lines in ${}^n{C}_2$ ways to get a point (say p).
Step 2nd : Now select another 2 lines in ${}^{n-2}{C}_2$ ways, to get another point (say Q)
Step 3rd : When P and Q are joined we get a fresh line.
But when we select P first then Q and Q first then P we get same line.
$\therefore \frac{{}^n{C}_2\times {}^{n-2}{C}_2}{2}$ Fresh lines

$=\frac{n(n-1)(n-2)(n-3)}{\mathrm{2.2.2}}$

$\therefore m+k=8+1=9$