Example 4. Team A and Team B each send out 7 players to participate in a Go competition according to a pre-arranged order. Both sides start with Player 1 competing, the loser is eliminated, and the winner then competes with the next (Player 2) of the losing side, $\cdots \cdots \cdots$, until one side is completely eliminated, and the other side wins, forming a competition process. How many different competition processes can occur? (1988, National High School Competition)
Let's first assume that Team A wins. Then, the 7 members of Team A need to eliminate the 7 members of Team B, which means they need to win a total of 7 games. This involves distributing these 7 games among the 7 members of Team A, with each distribution corresponding to a possible match process.

Thus, the problem is equivalent to finding the number of non-negative integer solutions to the equation $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}=7$. Using the method from Example 3, we find that the number of ways Team A can win is $C_{13}^{6}$.
Similarly, the number of ways Team B can win is also $C_{13}^{6}$.
Therefore, the total number of possible match processes is $2 C_{13}^{6}$.