Question

从图中网格点随机选取三个点，每组三点被选中的概率相同。这三个点在同一直线上的概率是多少？

A set of three points is chosen randomly from the grid shown. Each three-point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

$\dfrac{1}{21}$

$\dfrac{1}{14}$

$\dfrac{2}{21}$

$\dfrac{1}{7}$

$\dfrac{2}{7}$

答案：C

知识点：基本概率组合格点

难度评级：1240

小提示：

等可能的三点集合共有 $\binom{9}{3}$ 个。

There are $\binom{9}{3}$ equally likely three-point sets

大提示：

数共线三点： $3$ 行、 $3$ 列和 $2$ 条对角线。

Count the collinear triples: $3$ rows, $3$ columns, and $2$ diagonals

解答：

三点集合总数为 $\binom{9}{3} = 84\text{。}$

共线三点包括 $3$ 行、 $3$ 列和 $2$ 条主对角线，共 $8$ 组。

因此所求概率为 $\dfrac{8}{84} = \dfrac{2}{21}\text{。}$

所以正确答案是 C。

The number of three-point sets is $\binom{9}{3} = 84.$

The collinear triples are the $3$ rows, the $3$ columns, and the $2$ main diagonals, for a total of $8.$

The probability is therefore $\dfrac{8}{84} = \dfrac{2}{21}.$

Thus, the correct answer is C.

2004 AMC 10A 第 5 题

答案：C

难度评级：1240

小提示：

大提示：

解答：

其他年份的第 5 题