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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_17 / How many 4×4 arrays whose entries are 0s and 1s 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 …
Problem
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problem
How many arrays whose entries are s and s are there such that the row sums (the sum of the entries in each row) are and in some order, and the column sums (the sum of the entries in each column) are also and in some order? Output the remainder when the answer is divided by 100.
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.Plain-text mathematical notation (without MathML)
How many 4×4 arrays whose entries are 0s and 1s 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? Output the remainder when the answer is divided by 100.
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.Original LaTeX notation
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? Output the remainder when the answer is divided by 100.
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.Discussion
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