AMC12 2022 B

AMC12 2022 B · Q17

AMC12 2022 B · Q17. It mainly tests Basic counting (rules of product/sum), Casework.

How many $4 \times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1, 2, 3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1, 2, 3,$ and $4,$ in some order? For example, the array \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]\] satisfies the condition.

有且仅有 $1,2,3,4$（顺序任意）的行和，以及列和为 $1,2,3,4$（顺序任意）的 $4 \times 4$ 由 $0$ 和 $1$ 组成的阵列有多少个？例如阵列 \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]\] 满足条件。

(A) 144 144

(B) 240 240

(D) 576 576

(E) 624 624

Answer

Correct choice: (D)

正确答案：（D）

Solution

Note that the arrays and the sum configurations have one-to-one correspondence. Furthermore, the row sum configuration and the column sum configuration are independent of each other. Therefore, the answer is $(4!)^2=\boxed{\textbf{(D) }576}.$ Remark For any given sum configuration, we can uniquely reconstruct the array it represents. Conversely, for any array, it is clear that we can determine the unique sum configuration associated with it. Therefore, this establishes a one-to-one correspondence between the arrays and the sum configurations.

注意到阵列与其和配置一一对应。而且，行和配置与列和配置相互独立。因此答案为 $(4!)^2=\boxed{\textbf{(D) }576}$。备注对于给定的和配置，我们可以唯一地重构其代表的阵列。反之，对于任意阵列，显然可以确定与之关联的唯一和配置。因此，这建立了阵列与和配置之间的一一对应。

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