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Omni-MATH / In how many ways can 6 purple balls and 6 green balls be placed into a 4×4 grid of boxes such that every row and column contains two balls of one color and one ball of the other co…

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problem

In how many ways can 6 purple balls and 6 green balls be placed into a 4×44 \times 4 grid of boxes such that every row and column contains two balls of one color and one ball of the other color? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.
Plain-text mathematical notation (without MathML)
In how many ways can 6 purple balls and 6 green balls be placed into a 4×4 grid of boxes such that every row and column contains two balls of one color and one ball of the other color? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.
Original LaTeX notation
In how many ways can 6 purple balls and 6 green balls be placed into a $4 \times 4$ grid of boxes such that every row and column contains two balls of one color and one ball of the other color? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.

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