A chess Rook may travel in 470,010 ways from one corner square of a chessboard to the opposite corner square. A Queen may make the same trip in 1,499,858 ways.

A chess Rook may move any number of squares horizontally or vertically in one step. How many paths can a chess Rook take from the lower-left corner square to the upper-right corner square of an ordinary 8× 8 chessboard? Assume that the Rook moves right or up at each step. An example of such a Rook path is shown in Figure 9.1.