Question

In a polygon, no three diagonals are concurrent. If the total number of points of intersection of the diagonals interior to the polygon is 126, then number of diagonals of the polygon is

• A. $27$
• B. $28$
• C. $29$
• D. None of these

Answer 1

The correct option is A $27$

To find an internal intersection of 2 diagonals, we select 4 points from the vertices. These 4 points form a quadrilateral whose diagonals must intersect at a single point.
Thus, we have for each set of $4$ points, only $1$ point of intersection.
${}^n{C}_4=126=>n=9$ (by simple hit and trial)
Therefore, number of sides of the polygon $=9$ and number of diagonals ${=}^9{C}_2-9=36-9=27$
Hence, (A) is correct.