Editing Pascal's triangle (section)

==== 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 {{sqrt|2}} and one vertex at distance {{sqrt|3}} (the vertex opposite ''V''). The second row corresponds to a square, while larger-numbered rows correspond to [[hypercube]]s in each dimension.