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
Correct Answer: C
Solution :
Since no two lines are parallel and no three are concurrent, therefore \[n\] straight lines intersect at \[^{n}{{C}_{2}}=N\] (say) points. Since two points are required to determine a straight line, therefore the total number of lines obtained by joining \[N\] points \[^{N}{{C}_{2}}\]. But in this each old line has been counted \[^{n-1}{{C}_{2}}\] 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 \[^{N}{{C}_{2}}-n\ .{{\ }^{n-1}}{{C}_{2}}=\frac{N(N-1)}{2}-\frac{n(n-1)(n-2)}{2}\] \[=\frac{^{n}{{C}_{2}}{{(}^{n}}{{C}_{2}}-1)}{2}-\frac{n(n-1)(n-2)}{2}=\frac{n(n-1)(n-2)(n-3)}{8}\].
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