in reply to compute paths in Pascal's triangle (aka Tartaglia's one)

The property you mention is simple to prove: "the number in a specific tile is also the number of different shortest path from the top tile". In any path to a tile `n-k`, you have `(n-k)` (n minus k) moves left, and `k` moves right. Only the order of these moves determine the specific path. The number of possible permutations of (n-k) and k identical elements each is `n! / k! / (n-k)!` which is the number in the tile.

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