Question

在一个圆上选取八个点，并连接每一对点得到弦。没有三条弦在圆内部同一点相交。会形成多少个三个顶点都在圆内部的三角形？

Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?

$28$

$56$

$70$

$84$

$140$

答案：A

知识点：交点计数组合双射

难度评级：2160

解答：

一个内部三角形由三条在圆内部两两相交的弦形成。这样的三角形使用圆上的六个不同端点。

反过来，对任意选出的六个圆上点，按圆周顺序看，只有一种方法把相对端点配对成三条弦，使这三条弦在圆内两两相交。

因此三角形的数量为 $\binom{8}{6}=\binom{8}{2}=28$ 。

所以正确答案是 A。

An interior triangle is formed by three chords that pairwise intersect inside the circle. Such a triangle uses six distinct endpoints on the circle.

Conversely, for any six chosen points in circular order, exactly one set of three chords pairs opposite endpoints so that the three chords intersect pairwise inside the circle.

Therefore the number of triangles is $\binom{8}{6}=\binom{8}{2}=28.$

Thus, A is the correct answer.

2010 AMC 10A 第 22 题

答案：A

难度评级：2160

解答：

其他年份的第 22 题