Question

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 $\begin{bmatrix} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \end{bmatrix}$ satisfies the condition.

$144$

$240$

$336$

$576$

$624$

Solution:

The row with sum $4$ is all $1$ s and the column with sum $4$ is all $1$ s. There are $4!$ ways to assign the row sums $1, 2, 3, 4$ to the four rows, and $4$ choices for which column has sum $4.$

Delete that column. The remaining $4 \times 3$ array has row sums $0, 1, 2, 3$ and must have column sums $1, 2, 3.$ The all-zero and all-one rows are forced; the rows of reduced sum $1$ and $2$ can be placed in $6$ ways to produce column sums $1, 2, 3$ in some order.

The total is $24 \cdot 4 \cdot 6 = 576.$

Thus, the correct answer is D.

2022 AMC 12B Problem 17

Solution:

Problem 17 in Other Years