Question

数字 $1, 2, \ldots, 9$ 被随机放入一个 $3 \times 3$ 方格的 $9$ 个格子中。每个格子放一个数，每个数恰好使用一次。每一行和每一列中的数字和都为奇数的概率是多少？

The numbers $1, 2, \ldots, 9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

$\dfrac{1}{21}$

$\dfrac{1}{14}$

$\dfrac{5}{63}$

$\dfrac{2}{21}$

$\dfrac{1}{7}$

答案：B

知识点：奇偶性基本概率排列

难度评级：1800

解答：

有 $5$ 个奇数和 $4$ 个偶数。每一行和每一列都必须包含奇数个奇数项。

要放置 $5$ 个奇数项并使每一行和每一列的奇数项个数都为奇数，唯一的形状是填满一整行和一整列（一个由 $3 + 3 - 1 = 5$ 个格子组成的加号形）。这样的图案有 $3 \cdot 3 = 9$ 种。

每种图案中，奇数有 $5!$ 种放法，偶数有 $4!$ 种放法，所以概率为 $\dfrac{9 \cdot 5! \cdot 4!}{9!} = \dfrac{1}{14}.$

所以正确答案是 B。

There are $5$ odd and $4$ even numbers. Each row and column must contain an odd number of odd entries.

The only way to place $5$ odd entries with every row and column odd is to fill one complete row and one complete column (a plus shape of $3 + 3 - 1 = 5$ cells). There are $3 \cdot 3 = 9$ such patterns.

Each pattern admits $5!$ placements of the odd numbers and $4!$ of the even numbers, so the probability is $\dfrac{9 \cdot 5! \cdot 4!}{9!} = \dfrac{1}{14}.$

Thus, the correct answer is B.

2019 AMC 12A 第 16 题

答案：B

难度评级：1800

解答：

其他年份的第 16 题