Question

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

Answer 1

The number of points of intersection of $37$ lines is $37C_2$.But $13$ straight lines out of the given $37$ straight lines pass through the same point $A$.

Therefore instead of getting ${}^{13}{C}_2$points,we get only one point A.Similarly $11$ straight lines out of the given $37$ straight lines intersect at point $B$.Therefore instead of getting $11C_2$ points,we get only one point $B$.

Hence the number of intersection points of the lines is${}^{37}{C}_2-{}^{13}{C}_2-{}^{11}{C}_2+2=535$