When $x_1\ge 2$, this equation has no no non-negative integer solution.

When $x_1=0$, the original equation is equivalent to $$x_2+x_3+\cdots + x_9+x_{10}=3$$

which has $\binom{3+9-1}{9-1}=165$

When $x_1=1$, the original equation is equivalent to $$x_2+x_3+\cdots + x_9+x_{10}=1$$

which has $9$ non-negative integer solutions.

Therefore, the final answer is $$165+9=\boxed{174}$$