1999 AMC 12 第 24 题

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：

\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：

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