Question

从一个正八边形中随机选择三个顶点并连接成一个三角形。这个三角形至少有一条边也是八边形的一条边的概率是多少？

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

$\dfrac{2}{7}$

$\dfrac{5}{42}$

$\dfrac{11}{14}$

$\dfrac{5}{7}$

$\dfrac{6}{7}$

答案：D

知识点：补集计数隔板法

难度评级：1650

视频讲解：

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文字解答：

用补集计数：计算任意两个所选顶点都不相邻的三角形。从一个所选顶点出发，设三个间隔中未选顶点的正整数个数为 $x,y,z$ 。那么 $x+y+z=5$ ，共有 $\binom42=6$ 个正整数解。

起点有 $8$ 种选择，而每个三角形会从它的 $3$ 个顶点各被计算一次，所以补集中有 $8\cdot6/3=16$ 个三角形。全部三角形共有 $\binom83=56$ 个，因此所求概率是 $1-16/56=5/7$ 。

所以正确答案是 D。

Count the complement: triangles with no two chosen vertices adjacent. Starting at a chosen vertex, let $x,y,z$ be the positive numbers of unchosen vertices in the three gaps. Then $x+y+z=5$ , which has $\binom42=6$ positive solutions.

There are $8$ choices for the starting vertex, and each triangle is counted from each of its $3$ vertices, so the complement contains $8\cdot6/3=16$ triangles. Out of $\binom83=56$ total triangles, the desired probability is $1-16/56=5/7.$

Thus, D is the correct answer.

2018 AMC 8 第 23 题

答案：D

难度评级：1650

视频讲解：

文字解答：

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