All roads lead to Rome

Discrete Mathematics Level 3

Consider a chessboard. Those that might not be familiar with chess, a (classic) chessboard is made out of 64 squares (8 rows and 8 columns).

Suppose that the board is clear from any piece, except from the king, which you place at the lowest left corner (the red square A1). You goal is to move around the king to get to the upper right corner (the red square H8), but there are rules: you are only allowed to move either to the right or up (and hence, never diagonally or to the left or down).

How many different paths are there that satisfy those constraints and travel from A1 to H8?

Bonus : can you generalize to an \(m \times n\) chessboard?


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