Question

掷出 $7$ 个公平的标准 $6$ 面骰子，朝上点数之和为 $10$ 的概率可写成其中 $n$ 是正整数。求 $n$ 。 $\dfrac{n}{6^{7}},$

When $7$ fair standard $6$ -sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\dfrac{n}{6^{7}},$ where $n$ is a positive integer. What is $n?$

$42$

$49$

$56$

$63$

$84$

答案：E

知识点：骰子（概率）隔板法

难度评级：1420

解答：

用隔板法求 $n$ ：这等价于把 $10$ 个球放入 $7$ 个盒子，并要求每个盒子至少有一个球。

若球数为 $n$ ，盒子数为 $k$ ，这种放法的数量为。 $\binom{n - 1}{k - 1},$

因此所求数量为 $4$ $10$ $6$ $\binom{9}{6} = \binom{9}{3} = 84.$

因此正确答案是 E。

We can use stars and bars to find $n.$ It is the same as finding the number of ways to put $10$ balls into $7$ boxes, where each box has at least one ball.

The formula for such a scenario is $\binom{n - 1}{k - 1},$ where $n$ is the number of balls and $k$ is the number of boxes.

No die can exceed $4$ in a sum of $10$ from seven positive rolls, so the upper bound of $6$ creates no additional restriction. The desired answer is therefore $\binom{9}{6} = \binom{9}{3} = 84.$

Thus, E is the correct answer.

2018 AMC 10A 第 11 题

答案：E

难度评级：1420

解答：

其他年份的第 11 题