Example 11 A convex $n$-sided polygon, with no three diagonals concurrent, can form how many triangles with its sides and diagonals?
1. Let $T_{0}$ denote the set of triangles whose vertices are all intersection points (internal points) of the diagonals of a polygon. Suppose $\triangle A_{1} A_{2} A_{3} \in T_{0}$, then the three sides of this triangle lie on three diagonals of the convex $n$-sided polygon, which corresponds to a convex hexagon $B_{1} B_{2} B_{3} B_{4} B_{5} B_{6}$. Let the number of ways to choose 6 points from $n$ given points be denoted as $T^{\prime}_{0}$. Then, the 6 points $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}$ determine a selection in $T^{\prime}_{0}$. Conversely, for these 6 points, only connecting $B_{1} B_{4}$, $B_{2} B_{5}$, and $B_{3} B_{6}$ can yield a triangle in $T_{0}$, and any 6 points chosen from the $n$ vertices of the convex $n$-sided polygon can be connected in this manner to form a triangle in $T_{0}$. Thus, the elements of set $T_{0}$ can be put into a one-to-one correspondence with the elements of set $T^{\prime}_{0}$, so we have
   $$
   \left|T_{0}\right|=\left|T^{\prime}_{0}\right|=C_{n}^{6}.
   $$

2. Let $T_{1}$ denote the set of triangles that have exactly 1 vertex as a vertex of the convex $n$-sided polygon. By a similar argument as in 1, each 5 triangles in $T_{1}$ correspond to a 5-element subset of the vertices of the convex $n$-sided polygon. Conversely, different 5-element subsets correspond to 5 completely different triangles in $T_{1}$. Therefore, $\left|T_{1}\right|=5 C_{n}^{5}$.

3. Let $T_{2}$ denote the set of triangles that have exactly 2 vertices as vertices of the convex $n$-sided polygon. It can be seen that each 4 triangles in $T_{2}$ correspond to a 4-element subset of the vertices of the convex $n$-sided polygon. Conversely, different 4-element subsets correspond to 4 completely different triangles in $T_{2}$. Therefore, $\left|T_{2}\right|=4 C_{n}^{4}$.

4. Let $T_{3}$ denote the set of triangles that have exactly 3 vertices as vertices of the convex $n$-sided polygon. Clearly, $\left|T_{3}\right|=C_{n}^{3}$.

In summary, the union of $T_{0}, T_{1}, T_{2}, T_{3}$ is the set we are looking for, and the intersection of any two sets is empty. Therefore, the number of triangles that meet the conditions is
$$
|T|=\left|T_{0}\right|+\left|T_{1}\right|+\left|T_{2}\right|+\left|T_{3}\right|=C_{n}^{6}+5 C_{n}^{5}+4 C_{n}^{4}+C_{n}^{3}.
$$