2. Given the indeterminate equation $x+2 y+3 z=15$,
(1) Find the number of non-negative integer solutions;
(2) Find the number of positive integer solutions.
2.(1) From the problem, we have the generating function $f(x)=\frac{1}{(1-x)\left(1-x^{2}\right)\left(1-x^{3}\right)}$,

and $f(x)=\frac{1}{(1-x)^{3}} \cdot \frac{1}{1+x} \cdot \frac{1}{1+x+x^{2}}$
$$
=\frac{A}{1+x}+\frac{B}{1-x}+\frac{C}{(1-x)^{2}}+\frac{D}{(1-x)^{3}}+\frac{E}{1-\omega x}+\frac{F}{1-\omega^{2} x} \text {. }
$$

where $\omega=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}$, and by calculating the undetermined coefficients, we get
$$
A=\frac{1}{8}, B=\frac{17}{72}, C=\frac{1}{4}, D=\frac{1}{6}, E=\frac{1}{9}, F=\frac{1}{9} .
$$

Thus, $f(x)=\frac{1}{8(1+x)}+\frac{17}{72(1-x)}+\frac{1}{4(1-x)^{2}}+\frac{1}{6(1-x)^{3}}+\frac{1}{9(1-\omega x)}+\frac{1}{9\left(1-\omega^{2} x\right)}$
$$
\left.=\sum_{r=0}^{\infty}\left[\frac{1}{8}(-1)^{r}\right)+\frac{17}{72}+\frac{1}{4} C_{r+1}^{r}+\frac{1}{6} C_{r+2}^{r}+\frac{1}{9}\left(\omega^{r}+\omega^{2 r}\right)\right] x^{r}
$$

Therefore, $a_{r}=\frac{(r+2)(r+1)}{12}+\frac{r+1}{4}+\frac{17}{72}+\frac{1}{8}(-1)^{r}+\frac{2}{9} \cos \frac{2 r \pi}{3}$. Hence, $a_{15}=\frac{17 \times 16}{12}+\frac{16}{4}+\frac{17}{72}-\frac{1}{8}+\frac{2}{9}=27$ is the desired result.
(2) Let $x=x^{\prime}+1, y=y^{\prime}+1, z=z^{\prime}+1$, then the problem is transformed into finding the non-negative integer solutions of $x^{\prime}+2 y^{\prime}+3 z^{\prime}=9$. By (*) we get $a_{r}=12$, which is the required result.