{"status": "success", "data": {"description_md": "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\n\n$$\\left[
\\begin{array}{cccc}
1 & 1 & 1 & 0 \\\\
0 & 1 & 1 & 0 \\\\
1 & 1 & 1 & 1 \\\\
0 & 1 & 0 & 0 \\\\
\\end{array}
\\right]$$
satisfies the condition.\n\n$\\textbf{(A) }144 \\qquad \\textbf{(B) }240 \\qquad \\textbf{(C) }336 \\qquad \\textbf{(D) }576 \\qquad \\textbf{(E) }624$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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.

\\textbf{(A) }144 \\qquad \\textbf{(B) }240 \\qquad \\textbf{(C) }336 \\qquad \\textbf{(D) }576 \\qquad \\textbf{(E) }624

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2022 AMC 12B Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12B_p18", "prev": "/problem/22_amc12B_p16"}}