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?

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.