Chessmaster1000: AbigailII: there's at least one game with a position that repeats itself.

Can you explain in more details this.........

Eh, you have a game, and the positions (that is, where the pieces are on the board, and whose move it is, and if you have a cube, the value of the cube) after move N and M (for N and M not equal to each other) are the same. If that can happen in a game, you have an infinite amount of different backgammon games (if the position after move N can occur after move M again, it can also occur after move 2M - N, 3M - 2N, 4M - 3N, ..., kM - (k-1)N, k >= 0).

If positions cannot be repeated, the number of different games is finite - as the number of different positions is finite.