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Combinatorics Difficulty 4.8 AIME Prove it United States

Problem:

Consider a $2 \times 3$ grid where each entry is one of $0$ , $1$ , and $2$ . For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$ ? One valid grid is shown below.
$\left[\begin{array}{lll} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array}\right]$

Solution

Solution:

Any two elements in the same row fix the rest of the grid, so $3^{2} = 9$ .

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