A computer science teacher poses his students a problem.
"I want you write a computer program that plays tic-tac-toe
legally and runs through ALL the possible combinations of the game, and finds out the total."
The students settle down to work..
An hour later, a student gets up and proclaims "I've got it! The number of possible combinations in a game is 344,242."
At which point another student quickly replies, "I haven't finished yet, but I'm sure
Fred made a mistake in his program."
(Tic Tac Toe = Noughts and Crosses)
This article by Steve Shaffer on mathrec.org
discusses various ways to count and arrives at a few different totals:
255,168 different sequences of moves
31,896 different sequences after eliminating reflections and rotations of the final positions
26,830 possible games if two games are the same when the players faced the same choices at each move of the game (i.e., considering symmetry at each step of the game.)
23,129 possible games under the same symmetry restrictions, if the players quit when the outcome is determined.