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Problem 240

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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 $70$?

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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：

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

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？

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