12. The expansion of $(a+b+c)^{10}$, after combining like terms, has $\qquad$ terms.
12. 66

The form of each term in the combined store is $a^{K_{1}} b^{K_{2}} c^{K_{3}}$,
where $K_{1}+K_{2}+K_{3}=10,\left(K_{1} 、 K_{2} 、 K_{3} \in \mathbf{N}\right)$, 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, and each way corresponds to a like term, hence there are 66 terms.