Top Dice

There are $1111$ ways in which five $6$-sided dice (sides numbered $1$ to $6$) can be rolled so that the top three sum to $15$. Some examples are:

$$ \begin{equation} \begin{aligned} &&D_1,D_2,D_3,D_4,D_5 &= 4,3,6,3,5\\\ &&D_1,D_2,D_3,D_4,D_5 &= 4,3,3,5,6\\\ &&D_1,D_2,D_3,D_4,D_5 &= 3,3,3,6,6\\\ &&D_1,D_2,D_3,D_4,D_5 &= 6,6,3,3,3\\\ \end{aligned} \notag \end{equation} $$

In how many ways can twenty $12$-sided dice (sides numbered $1$ to $12$) be rolled so that the top ten sum to $70$?