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:

$D_{1}, D_{2}, D_{3}, D_{4}, D_{5} = 4, 3, 6, 3, 5$
$D_{1}, D_{2}, D_{3}, D_{4}, D_{5} = 4, 3, 3, 5, 6$
$D_{1}, D_{2}, D_{3}, D_{4}, D_{5} = 3, 3, 3, 6, 6$
$D_{1}, D_{2}, D_{3}, D_{4}, D_{5} = 6, 6, 3, 3, 3$

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

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.