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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
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
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Solution
The correct option is C n(n−1)(n−2)(n−3)8
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 = N(N−1)2−n(n−1)(n−2)2
= nC2(nC2−1)2−n(n−1)(n−2)2 = n(n−1)(n−2)(n−3)8
n(n−1)(n−2)(n−3)8
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 = N(N−1)2−n(n−1)(n−2)2
= nC2(nC2−1)2−n(n−1)(n−2)2 = n(n−1)(n−2)(n−3)8
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