Example 4: Team A and Team B each send out 7 members to participate in a Go chess tournament according to a pre-arranged order. Both sides start with the No. 1 member competing, the loser is eliminated, and 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 wins, forming a competition process. Therefore, the number of all possible competition processes is


Let the members of Team A be denoted as $A_{1}, A_{2}, \cdots, A_{7}$, and the members of Team B as $B_{1}, B_{2}, \cdots, B_{7}$, with the subscripts indicating their predetermined order of appearance. A match process corresponds to the "ordered" permutation of these 14 elements, where the order from left to right represents the order in which players are eliminated, and the last element is the winner who has not been defeated or has not participated. By selecting 7 positions out of these 14 to sequentially arrange $A_{1}, A_{2}, \cdots, A_{7}$, there are $\mathrm{C}_{14}^{7}$ ways to arrange them. The remaining 7 positions are then sequentially arranged with $B_{1}, B_{2}, \cdots, B_{7}$, which has 1 way of arrangement. Therefore, the "ordered" (meaning the subscripts increase) permutation of the 14 elements has $\mathrm{C}_{14}^{7}$ ways, meaning there are $C_{14}^{7}$ possible match processes (see reference [9]).

It is evident that $\mathrm{C}_{14}^{7}$ reflects the structure of the problem and more fundamentally, the essential relationship.

In general, when Team A has $m$ members and Team B has $n$ members $\left(m, n \in \mathbf{N}_{+}\right)$, the match process is $\mathrm{C}_{m+n}^{m}$ or $\mathrm{C}_{m+n}^{n}$. The above explanation remains valid.

And $\mathrm{C}_{m+n}^{m}=\mathrm{C}_{m+n-1}^{n}+\mathrm{C}_{m+n-1}^{m}$ can represent that there are $\mathrm{C}_{m+n-1}^{n}$ ways for Team A to win, plus $\mathrm{C}_{m+n-1}^{m}$ ways for Team B to win. This situation can also be illustrated by a "grid route diagram" with $m$ columns and $n$ rows (as shown in Figure 3), representing all different paths from endpoint $A$ to endpoint $B$ by moving up or to the right. Each move to the right (up) represents the elimination of one member from Team A (B). Finally, if the path from left to right to point $B$ represents Team A's victory; if the path from bottom to top to point $B$ represents Team B's victory. Figure 3 can truly reflect the deeper structure of the problem.