2018 AMC 8 第 23 题
先试着解答 2018 AMC 8 第 23 题，然后核对你的答案与视频与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2018 AMC 8 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
23.
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?
小提示：
使用补集：数没有相邻顶点被选中的三角形
Use complementary counting: count triangles with no adjacent chosen vertices.
大提示：
固定一个顶点后，数三个选中顶点之间的正间隔
After fixing one vertex, count the positive gaps around the octagon between the chosen vertices.
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 。那么 ，共有 个正整数解。
起点有 种选择，而每个三角形会从它的 个顶点各被计算一次，所以补集中有 个三角形。全部三角形共有 个，因此所求概率是 。
所以正确答案是 D。
Count the complement: triangles with no two chosen vertices adjacent. Starting at a chosen vertex, let be the positive numbers of unchosen vertices in the three gaps. Then , which has positive solutions.
There are choices for the starting vertex, and each triangle is counted from each of its vertices, so the complement contains triangles. Out of total triangles, the desired probability is
Thus, D is the correct answer.
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