AMC 8 2018 Problem 23

AMC8 2018 年第 23 题

概率★★★★☆

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?

如下图所示的正八边形中，从 8 个顶点中任选 3 个并连接起来形成一个三角形。那么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？

A. $\displaystyle \frac{2}{7}$
B. $\displaystyle \frac{5}{42}$
C. $\displaystyle \frac{11}{14}$
D. $\displaystyle \frac{5}{7}$
E. $\displaystyle \frac{6}{7}$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 正面分类共

\binom83=56

个三角形。看至少共一边的：恰共 1 边——选边 8 种 × 第三顶点（不与该边两端相邻，余 4）=32；共 2 边——三个连续顶点，8 个。合计 40，概率

40/56=\tfrac57

。

Solution 1

Elementary · count directly There are

\binom83=56

triangles in all. Count those sharing at least one side with the octagon: exactly one shared side — pick the side (8 ways) and a third vertex not adjacent to its endpoints (4 left), giving 32; two shared sides — three consecutive vertices, 8. That is 40, so the probability is

40/56=\tfrac57

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