Tic-Tac-Toe Game What is the total number of strategies (pure strategies in game theory) in a Tic Tac Toe game for each player? Assume that  it is a 3x3 game and 2 players. 
Rule: The players put marks such as X and O sequentially. Both players observe all choices. The first player to have 3 of his/her marks in a row (horizontally, vertically or diagonally) wins. 
I was looking at this post (A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe) but it is a specific case that only considers winning strategies. 
My solution strategy:
I know that there are 9! ways to play this game for one player, which is 362880. BUT this is the case if the game does not end in 5th, 6th or other moves. I am not really sure how to add in these factors.
 A: Let's assume you mean pure strategies, ignore for the moment the fact that the game can end early, and not impose consistency on "transpositions"... i.e., let a player respond to a state differently depending on how it was arrived at.  The first player has nine possible first moves, so he has $N_1=9$ pure strategies for his first move.  The second player can respond in eight different ways, and the first player has seven possible responses to each of the eight situations he could find himself in on his second move.  He can choose these responses independently, so he has $N_2=9\cdot 7^8$ pure strategies for his first two moves.  He could find himself in any of $8\cdot 6=48$ situations on his third move, and can respond to each in five different ways, so he has $N_3=9\cdot 7^8 \cdot 5^{8\cdot 6}$ strategies for his first three moves.  Continuing, we find
$$
N = 9\cdot 7^8 \cdot 5^{8\cdot 6}\cdot 3^{8\cdot 6\cdot 4}\approx 7.46\times 10^{132}
$$
strategies for the first player for the entire game.  The second player, on the other hand, has
$$
N'=8^9 \cdot 6^{9\cdot 7}\cdot 4^{9\cdot 7 \cdot 5}\cdot 2^{9\cdot 7 \cdot 5 \cdot 3}\approx 1.88\times 10^{531}
$$
strategies.  Both these numbers are exponentially larger than either the number of positions or the number of games (and would remain so even if consistency were enforced and early endings were handled) but this is as it should be.  A strategy, after all, is a map from positions that you might see to moves that you would make in those positions.  As such, the number of strategies should be something like a typical branching factor to the power of the number of reachable positions.
