【Example 4】To place $k$ indistinguishable red balls and $k$ indistinguishable white balls into $n$ different boxes, each box can only hold balls of one color. If placing red balls, each box is limited to one, if placing white balls, the quantity is unlimited. How many different ways are there to place the balls $(1 \leqslant k \leqslant n-1)$?
Since at most one red ball can be placed in a box, the number of ways to place them is $C_{n}^{k}$. The white balls should be placed in the remaining $n-k$ boxes, which is a combination with repetition, i.e., $C_{n-k+(k-1)}^{k}=C_{n-1}^{k}$ ways. Therefore, the total number of ways to place the balls is $C_{n}^{k} \cdot C_{n-1}^{k}$.