Example 1: Team A and Team B each send out 7 members to participate in a Go team tournament according to a pre-arranged order. Both sides start with the No. 1 member competing, the loser is eliminated; the winner then competes with the No. 2 member of the losing side, ... until all members of one side are eliminated, and the other side is declared the winner, forming a competition process. How many different possible competition processes are there?
Solution: Since the losing side is completely eliminated, the total number of eliminations is 7. Let's assume side A wins, and let the $i$-th member of side A eliminate $x_{i}$ members of side B. Then the problem is equivalent to finding the number of non-negative integer solutions to the equation
$$
x_{1}+x_{2}+\cdots+x_{7}=7
$$

which has $\mathrm{C}_{13}^{6}$ solutions.
Similarly, if side B wins, there are also $\mathrm{C}_{13}^{6}$ solutions.
Therefore, the total number of possible match processes is $2 \mathrm{C}_{13}^{6}=3432$.

Thought: If side A only used its first 5 members to defeat side B's 7 members, how many possible match processes are there?