Question

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

• A. $426$
• B. $535$
• C. $601$
• D. $728$

Answer 1

The correct option is B $535$

The number of points of intersection of
$37$ straight lines is${}^{37}{C}_2$, but $13$ of them pass through the point $A$.
Therefore, instead of getting ${}^{13}{C}_2$ points we get merely one point.
Similary, $11$ straight lines out of the given $37$ straight lines intersect at $B$.
Therefore, instead of getting ${}^{11}{C}_2$ points, we get only one point.
Hence, the number of intersection points of the lines is
$={}^{37}{C}_2-{}^{13}{C}_2-{}^{11}{C}_2+2$
$=535$