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. $\frac{2}{7}$

B. $\frac{5}{42}$

C. $\frac{11}{14}$

D. $\frac{5}{7}$

E. $\frac{6}{7}$

1. 2018 AMC 8 官方解析

method 1 :

We consider two different scenarios: the triangle contains three consecutive vertices or two consecutive vertices. In the first scenario, there are eight ways to form a triangle by connecting three consecutive vertices. In the second scenario, there are $8 \times 4$ ways to form a triangle by connecting two consecutive vertices and another non-consecutive vertex. More specifically, 8 represents the number of ways to pick two consecutive vertices, and 4 represents the number of ways to pick another non-consecutive vertex. In total, there are $\binom{8}{3} = 8 \times 7$ ways to form a triangle. The probability can be calculated as follows:
$\frac{8 + 8 \times 4}{8 \times 7} = \frac{1 + 4}{7} = \frac{5}{7}$.

method 2 :

There are eight ways to pick two consecutive vertices and six ways to pick another vertex. However, note that the cases of all three vertices being consecutive are counted twice. We therefore subtract the number of ways to pick three consecutive vertices, which is eight. The probability can be calculated as follows: $\frac{8 \times 6 - 8}{8 \times 7} = \frac{5}{7}$. The correct answer is D.