Example 2: In the China-Japan Go Match, both sides send out 6 players to compete in a predetermined order until one side wins. How many possible match processes are there?
Solution: Let the Chinese players correspond to red balls, and the Japanese players correspond to white balls. Arrange the balls corresponding to the eliminated players one by one, with the balls of the players eliminated earlier placed in front. When all players of one side are eliminated, the corresponding balls are all placed, and then the balls corresponding to the remaining players of the other side are placed in sequence. Since the order of the players' appearances is fixed, we can assume that balls of the same color are indistinguishable. Thus, a game process corresponds to a permutation of 6 red balls and 6 white balls; conversely, a permutation of 6 red balls and 6 white balls corresponds to a game process. That is, a 1-1 correspondence is established between different permutations of the balls and different game processes. The number of different permutations of 6 red balls and 6 white balls is $C_{12}^{6}$.