3. Arrange $r$ boxes in a row, and put all $n$ identical balls into these arranged boxes. How many different ways are there to do this?
3. There are $r$ boxes numbered as 1, 2, $\cdots \cdots, r$, and the number of balls in them are $x_{1}, x_{2}, \cdots, x_{r}$, respectively, such that $x_{1}+x_{2}+\cdots+x_{r}=n$, and $x_{i} \geqslant 0(i=1,2,3, \cdots, r)$. Let $x_{i}^{\prime}=x_{i}+1$ $\geqslant 1(i=1,2, \cdots \cdots, r)$, then the original equation becomes $x_{1}{ }^{\prime}+x_{2}{ }^{\prime}+\cdots \cdots x_{r}{ }^{\prime}=n+r$, where $x_{i}^{\prime}$ $\geqslant 1(i=1,2,3, \cdots \cdots, r)$. In summary, the different ways to place the balls correspond to the non-negative integer solutions of $x_{1}+x_{2}+\cdots \cdots+x_{r}$ $=n$, and its non-negative integer solutions are in 1-1 correspondence with the positive integer solutions of $x_{1}{ }^{\prime}+x_{2}{ }^{\prime}+x_{3}{ }^{\prime}+\cdots \cdots+x_{r}{ }^{\prime}=n+r$. The positive integer solutions of $x_{1}{ }^{\prime}+x_{2}{ }^{\prime}+\cdots \cdots+x_{\mathrm{r}}{ }^{\prime}=n+r$ are $C_{n+r-1}^{r-1}$, so the different ways to place the balls are also $C_{n+r-1}^{r-1}$.