There are $p\ge6$ points given on circumference of a circle, and every two points are joined by a chord. Assume that no 3 chords are concurrent within the interior of the circle. We have to find the number of intersection points made by these chords inside the circle.
We count the ways to choose 4 points out of $p$ and each set of the chosen points contributes to a single intersection point inside the circle. So the answer is $$p\choose4$$ But I cannot understand the significance of the condition in bold. How will the answer change without this condition?