Example 1 If in a convex $n$-sided polygon no three diagonals are concurrent, find the number of regions into which the interior of the $n$-sided polygon is divided by these diagonals.

The method of successive elimination, also known as the Bonsa method, is commonly used by graph theorist Bonsa. This approach considers the problem from the opposite angle, i.e., gradually removing all the diagonals, each time removing one and analyzing the reduction in regions during this process.

Removing one diagonal at a time, while analyzing the reduction in regions during this process.
Consider, as shown in Figure 1, the number of regions that disappear due to the cancellation of the diagonal \( A_{k} A_{j} \). When this diagonal is gradually lifted (imagine it being erased from one end, segment by segment), each segment that is lifted merges the two regions it previously divided. Since the number of line segments equals the number of internal intersection points plus 1, and only two diagonals pass through each internal intersection point, the cancellation of one of these diagonals will also remove this intersection point from the other diagonal. Therefore, when it comes to canceling a subsequent diagonal, many of its original intersection points, or even all of them, may have already been removed. Thus, when a diagonal is canceled, the number of regions that decrease is the number of intersection points remaining on this diagonal before its cancellation plus 1 (not the initial number of intersection points plus 1).

Furthermore, note that during the entire process of canceling the diagonals, the number of regions that decrease is recorded sequentially, and each intersection point is counted exactly once throughout the process. Therefore, the total number of regions that disappear in this process is:
(Intersection points on the first diagonal + 1) +
(Intersection points remaining on the second diagonal + 1) \( + \cdots + \)
(Intersection points remaining on the last diagonal + 1)
\( = \) (Total number of internal intersection points) + (Number of diagonals) \( \times 1 \)
\( = \mathrm{C}_{n}^{4} + (\mathrm{C}_{n}^{2} - n) \).

At the same time, after all diagonals are removed, one region should remain (i.e., the \( n \)-sided polygon itself). Therefore, the original number of regions is
\[
f(n) = \mathrm{C}_{n}^{4} + (\mathrm{C}_{n}^{2} - n) + 1 = \mathrm{C}_{n}^{4} + \mathrm{C}_{n-1}^{2}.
\]