Lines $L_1,L_2,...,L_{100} $ are distinct .All lines $L_{4n}$ ,$n$ a positive integer, are parallel to each other. All lines $L_{4n-3}$, $n$ a positive integer,pass through a given point $A$. Find the maximum number of point of intersection of pairs of lines from the complete set $ {L_1,L_2,...,L_{100} }$.


$a) 4350$

$b) 4351$

$c) 4900$

$d) 4901 $

My attempt:

I've calculated the number of points of intersection of pairs of lines with none of them parallel to some other line,so I had $\dbinom {100}{2} $.

From this I have subtracted the pairs of lines which are parallel to each other,yelding $\dbinom {100}{2} - \dbinom {25}{2} = 4650 $ .

But this is wrong if I look at the options given.Where's my mystake ?

Each non-parallel and parallel line will have $1$ point of intersection,isn't it ?

  • Altogether, $100$ lines have at most ${100\choose2}=4950$ intersection points.
  • You lose all the ${25\choose2}=300$ intersection points of the $25$ parallel lines.
  • You lose all but one of the ${25\choose2}=300$ intersection points of the lines through point $A$.

This yields a maximum number of $4950-300-299=4351$ intersection points.
Hence answer (b) is correct.

  • $\begingroup$ Why do I lose $300$ points of intersection of the lines through $A$ ?We count all these intersections as $1$ since they pass through the same point ? $\endgroup$ – Nameless Nov 20 '15 at 10:37
  • $\begingroup$ @Nameless: Yes. $\endgroup$ – Gamow Nov 20 '15 at 10:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.