12. The expansion of $(a+b+c)^{10}$, after combining like terms, has
$\qquad$ terms.
12.66.

The form of each term after merging is $a^{k_{1}} b^{k_{2}} c^{k_{3}}$, where $k_{1}+k_{2}+k_{3}=10\left(k_{1} 、 k_{2} 、 k_{3} \in \mathrm{N}\right)$, and when $k_{1} 、 k_{2}$ are determined, $k_{3}$ is uniquely determined.
If $k_{1}=10, k_{2}=0$, there is 1 way;
If $k_{1}=9, k_{2}=0$ or 1, there are 2 ways;
If $k_{1}=8, k_{2}=0 、 1 、 2$, there are 3 ways;
......
If $k_{1}=1, k_{2}=0 、 1 、 2 、 \cdots 、 9$, there are 10 ways;
If $k_{1}=0, k_{2}=0 、 1 、 2 、 \cdots 、 10$, there are 11 ways.
In total, there are $1+2+3+\cdots+10+11=66$ ways, each

way corresponds to one like term, hence there are 66 terms.