structural:expansion, structural:reformatting, information:filtering, linguistic:voice_change
Counting vertices in a cube by distance

Each row of Pascal's triangle gives the number of vertices at each distance from a fixed vertex in an n -dimensional cube. For example, in three dimensions, the third row (1 3 3 1) corresponds to the usual three-dimensional cube: fixing a vertex V, there is one vertex at distance 0 from V (that is, V itself), three vertices at distance 1, three vertices at distance √ 2 and one vertex at distance √ 3 (the vertex opposite V). The second row corresponds to a square, while larger-numbered rows correspond to hypercubes in each dimension.