Straight lines in combinatorics!
Probability
Level
3
There are 12 straight lines in a plane and no two of them are parallel and no three pass through the same point. Their point of intersections are joined .Find the no. of fresh lines thus introduced!
2880
2000
1485
765
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1 solution
Let AB be any one of the line and suppose it is intersected by another line CD at some pt P.
It is clear that AB contains 11 pts of intersections because it is intersected by remaining 11 lines at ll different pts.
Therefore (n-1) no. of points are contained in n lines. Here n=12.
Except the pt P no. of points contained in AB and CD =10
Therefore no. of pts situated on AB and CD=2(n-2)-1=2n-3=21 (n=12)
Number of pts outside AB and CD=(n(n-1)/2)-(2n-3)=(W) The aggregate no. of new lines produced from these points =n(n-1)/2(W)
But in the above every line is counted twice ....
For instance if Q be one of those pts outside AB and CD line PQ is counted once among the lines passing through P and once among those passing through Q ...
Therefore no. of lines =0.5(n(n-1))/2 [W] =(1/8)(n(n-1)(n-2)(n-3)) =12(11) (10)*(9/8)=1485 lines.....