Problem 240

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:

$\begin{aligned} D_{1}, D_{2}, D_{3}, D_{4}, D_{5} & = 4, 3, 6, 3, 5 \\ D_{1}, D_{2}, D_{3}, D_{4}, D_{5} & = 4, 3, 3, 5, 6 \\ D_{1}, D_{2}, D_{3}, D_{4}, D_{5} & = 3, 3, 3, 6, 6 \\ D_{1}, D_{2}, D_{3}, D_{4}, D_{5} & = 6, 6, 3, 3, 3 \end{aligned}$

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

点数最大的骰子

扔出 $5$ 个六面骰子（面上分别标有 $1$ 至 $6$ ），其中三个最大的点数之和为 $15$ ，一共有 $1111$ 种方式。其中的一些例子为：

扔出 $20$ 个 $12$ 面骰子（面上分别标有 $1$ 至 $12$ ），其中十个最大的点数之和为 $70$ ，一共有多少种方式？