Out of all possible configurations of a regular $8\times 8$ chess board, how many configurations are impossible?

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Given the usual rules and initial configuration of pieces. There are some obvious ones that come to mind, such as any configuration with a pawn on 1 or 8, or all pawns on 2 and anything but a white knight on any square (analogous example for 7/black). Does anyone know of an elegant, combinatorial calculation, rather than a brute force computation?