A rectangle over the shaded row

HardProbability Combinatorics~9m

In a $2023 \times 2023$ grid of unit squares, the entire middle row (the $1012$ th) is shaded. A rectangle is chosen uniformly at random from all axis-aligned rectangles made of grid cells. The probability that it contains at least one shaded square equals $\tfrac{1012}{M}$ for an integer $M$ ; find $M$ .

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The shaded row spans all columns, so only the rectangle's vertical range matters: does it include the middle row?
Count line-pairs that fail to span it (both above or both below): $2\binom{1012}{2}/\binom{2024}{2}=1011/2023$ .

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A rectangle over the shaded row

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