Example 10 A regular $n$-sided polygon inscribed in a circle, with no three diagonals concurrent, try to find the number of quadrilaterals formed by these diagonals with vertices inside the circle?


Solution: From Figure $30-3$, we can see that any quadrilateral with vertices inside the circle corresponds to 8 points on the circumference; conversely, by connecting every second point among 8 points on the circumference, a quadrilateral with vertices inside the circle can certainly be formed. This indicates a 1-1 correspondence between the quadrilaterals with vertices inside the circle and the groups of 8 points on the circumference. There are $C_{n}^{8}$ such groups of 8 points. Therefore, the number of quadrilaterals that meet the condition is $C_{n}^{8}$.