Let there be $9$ fixed points on the circumference of a circle . Each of these points is joined to every one of the remaining $8$ points by a straight line and the points are so positioned on the circumference that atmost $2$ straight lines meet in any interior point of the circle. The number of such interior intersection points is :
My attempt is as follows:-
Let's find out how many line or line segments we can construct out of $9$ points, it will be $\displaystyle{9\choose 2}=36$
Now atmost two of these line segments can intersect at any interior point of the circle, so number of possible pairs of line segments which can meet in any interior point of the circle can be $\displaystyle{36\choose 2}=630$
So, number of such possible interior intersection points can be $630$
But actual answer is $\displaystyle{9\choose4}=126$. This also seems incorrect because if we select $4$ fixed points, then out of these $4$ fixed points we can construct $3$ pairs of line segments and each pair of line segment can produce $1$ interior intersection point. So no of such interior intersection points can be $\displaystyle{9\choose4}\cdot3=378$
Like if we select $A,B,C,D$, then following pairs of line segments can be constructed
$\{AB,CD\},\{AC,BD\},\{AD,BC\}$
What mistake am I making here?
I know this question has been asked before, but I was not finding any of the answers satisfactory. Please help me in this.