AMC8 2018

AMC8 2018 · Q23

AMC8 2018 · Q23. It mainly tests Combinations, Probability (basic).

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?

从一个正八边形中，通过连接随机选择的三个顶点形成一个三角形。三角形至少有一条边也是八边形的边的概率是多少？

(A) $\frac{2}{7}$ $\frac{2}{7}$

(B) $\frac{5}{42}$ $\frac{5}{42}$

(D) $\frac{5}{7}$ $\frac{5}{7}$

(E) $\frac{6}{7}$ $\frac{6}{7}$

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): For each side of the octagon, there are $6$ triangles containing that side. Because the $8$ triangles containing two adjacent sides of the octagon are counted twice, there are a total of $8\cdot6-8=40$ triangles sharing a side with the octagon. The total number of triangles that can be formed from the eight vertices is $\frac{8\cdot7\cdot6}{3!}=56$, so the probability is $\frac{40}{56}=\frac{5}{7}$.

答案（D）：对于八边形的每一条边，都有 $6$ 个包含该边的三角形。由于包含八边形两条相邻边的那 $8$ 个三角形被重复计算了两次，所以与八边形共享一条边的三角形总数为 $8\cdot6-8=40$。由这 8 个顶点能组成的三角形总数为 $\frac{8\cdot7\cdot6}{3!}=56$，因此所求概率为 $\frac{40}{56}=\frac{5}{7}$。

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