Count Rankings

A competition consists of 10 players $P_1, P_2, \dots, P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$ : the loser is ranked 10th, while the winner plays $P_8$ . The loser of that game is ranked 9th, while the winner plays $P_7$ . They keep repeating this process until someone plays $P_1$ : the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible?