Eccoci.

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:

$D_1,D_2,D_3,D_4,D_5 = 4,3,6,3,5$
$D_1,D_2,D_3,D_4,D_5 = 4,3,3,5,6$
$D_1,D_2,D_3,D_4,D_5 = 3,3,3,6,6$
$D_1,D_2,D_3,D_4,D_5 = 6,6,3,3,3$

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