(2) How many non-negative integer solutions does the equation $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ have?
The number 3 can be partitioned as
$$
3,1+2,2+1,1+1+1 .
$$
(i) For 3, we have
$$
\begin{array}{l}
x_{1}=0, x_{2}=3, x_{3}=\cdots=x_{10}=0 ; \\
x_{1}=0, x_{2}=0, x_{3}=3, x_{4}=\cdots=x_{10}=0 ; \\
\cdots \cdots \\
x_{1}=0, x_{2}=\cdots=x_{9}=0, x_{10}=3 ;
\end{array}
$$

There are $\mathrm{C}_{9}^{1}=9$ solutions.
(ii) For $1+2$, we have
$$
\begin{array}{l}
x_{1}=0, x_{2}=1, x_{3}=2, x_{4}=\cdots=x_{10}=0 ; \\
x_{1}=0, x_{2}=1, x_{3}=0, x_{4}=2, x_{5}=\cdots=x_{10}=0 ; \\
\cdots \cdots
\end{array}
$$

There are $\mathrm{A}_{9}^{2}=9 \cdot 8=72$ solutions.
(iii) For $2+1$, we have
$$
\begin{array}{l}
x_{1}=1, x_{2}=1, x_{3}=\cdots=x_{10}=0 ; \\
\cdots \cdots \\
x_{1}=1, x_{2}=0, x_{3}=\cdots=x_{9}=0, x_{10}=1 .
\end{array}
$$

There are $\mathrm{C}_{9}^{1}=9$ solutions.
(iv) For $1+1+1$, we have
$$
\begin{array}{l}
x_{1}=0, x_{2}=x_{3}=x_{4}=1, x_{5}=\cdots=x_{10}=0 ； \\
\cdots \cdots \\
x_{1}=0, x_{2}=\cdots=x_{7}=0, x_{8}=x_{9}=x_{10}=1 .
\end{array}
$$

There are $\mathrm{C}_{9}^{3}=\frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1}=3 \cdot 4 \cdot 7=84$ solutions.
Thus, the equation has
$$
\mathrm{C}_{9}^{1}+\mathrm{A}_{9}^{2}+\mathrm{C}_{9}^{1}+\mathrm{C}_{9}^{3}=9+72+9+84=174
$$

non-negative integer solutions.