Example 3: Team A and Team B each send out 7 players to participate in a Go team match according to a pre-arranged order. Both sides start with the No. 1 players competing. The loser is eliminated, and the winner then competes with the No. 2 player of the losing side... until all players of one side are eliminated. The other side wins. This forms a match process. How many different match processes can occur?
Team A players are represented by black balls, and Team B players are represented by white balls $\bigcirc$. Each match process in the tournament corresponds to a permutation of 7 black balls and 7 white balls, which can be verified as a one-to-one mapping. For example, the permutation
$0 \bigcirc 0 \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc 0 \bigcirc$

represents a match process: Team A's lead lost, the 2nd player of Team A defeated the 1st and 2nd players of Team B but lost to the 3rd player of Team B. The 3rd and 4th players of Team A lost consecutively, and the 5th player of Team A defeated the 3rd and 4th players of Team B but lost to the 5th player of Team B. The 6th player of Team A consecutively defeated the 5th, 6th, and 7th players of Team B. Team A won the match. The 7th player of Team A did not play. Conversely, each match process uniquely corresponds to a permutation of 7 black balls and 7 white balls, so there are a total of $C_{14}^{7}=3432$ different match processes.