Question

给定圆上的六个点。从连接这六点两两之间的弦中随机选择四条。四条弦构成一个凸四边形的概率是多少？

Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral?

$\dfrac{1}{15}$

$\dfrac{1}{91}$

$\dfrac{1}{273}$

$\dfrac{1}{455}$

$\dfrac{1}{1365}$

答案：B

知识点：基本概率组合

难度评级：1880

小提示：

从 $6$ 个圆上点中任选 $4$ 个，恰好确定一个凸四边形。

Any $4$ of the $6$ points determine exactly one convex quadrilateral

大提示：

先数从 $\binom{6}{2}$ 条弦中选 $4$ 条的所有方式。

Count all ways to choose $4$ chords from the $\binom{6}{2}$ chords

解答：

共有 $\binom{6}{2} = 15$ 条弦，因此选择四条弦的方法数为 $\binom{15}{4} = 1365$ 。四条弦形成凸四边形，当且仅当它们是某 $4$ 个圆上点所形成四边形的四条边。

任选四个点恰好给出一个这样的四边形，所以有利情况为 $\binom{6}{4} = 15$ ，概率为 $\dfrac{15}{1365} = \dfrac{1}{91}$ 。

所以正确答案是 B。

There are $\binom{6}{2} = 15$ chords, so $\binom{15}{4} = 1365$ ways to select four of them. A convex quadrilateral arises exactly when the four chords are the sides of a quadrilateral on four of the six points, and each choice of $4$ points gives exactly one such quadrilateral.

Hence there are $\binom{6}{4} = 15$ favorable outcomes, and the probability is $\dfrac{15}{1365} = \dfrac{1}{91}.$

Thus, the correct answer is B.

1999 AMC 12 第 24 题

答案：B

难度评级：1880

小提示：

大提示：

解答：

其他年份的第 24 题