【Example 2】To place $n+1$ distinct balls into $n$ distinct boxes, how many ways are there such that no box is empty?
To ensure no box is empty, only two of the balls must be placed in the same box, while the rest of the balls each go into a separate box. Therefore, there are two questions here: (1) Which two balls are placed in the same box? (2) How are all the balls placed into the boxes? Clearly, question (1) is a combination problem, with $C_{n}^{2}$ different ways. Question (2) is a permutation problem, involving placing $n$ objects (one of which is the pair of balls in the same box, and the rest are individual balls) into $n$ boxes, which can be done in $A_{n}^{n}=n!$ different ways. Therefore, there are a total of $C_{n}^{2} \cdot n!$ ways to ensure no box is empty.