Question

In a plane there are $37$ st. lines, of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides, no three lines pass through one point, no line passes through both points $A$ and $B$ and no two are parallel. Then the number of intersection points the lines have is equal to

• A. $535$
• B. $601$
• C. $728$
• D. none of these

Answer 1

Answer: (A)

The number of points of intersection of $37$ st. lines $=\,{}^{37}C_{2}$.
But $13$ of them pass thro’ the point $A$.
$\therefore $ instead of getting $^{37}C_{2}$ points, we get merely one point. Similarly $11$ st. lines out of the given $37$ st. lines intersect at $B$.
$\therefore $ instead of getting $^{37}C_{2}$ points, we get only one point. Hence the number of points of intersection the lines is
$^{37}C_{2}-\,{}^{13}C_{2}-\,{}^{11}C_{2}+2$
$=535$