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 $m + k$ ?

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Solution

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.
$∴ \frac{^{n} C_{2} \times^{n - 2} C_{2}}{2}$ Fresh lines
$= \frac{n (n - 1) (n - 2) (n - 3)}{2.2.2}$
$∴ m + k = 8 + 1 = 9$

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