Number of Ways Player A Wins Badminton Series

To determine the total number of ways in which player $A$ can win the badminton series, we analyze the condition required for player $A$ to be declared the winner.

Player $A$ wins the series as soon as $A$ achieves $5$ wins. Since no game ends in a draw, the series can last anywhere from $5$ to $9$ games.

For the series to end on the $n^{\text{th}}$ game with player $A$ winning:

1. Player $A$ must win the $n^{\text{th}}$ game.
2. Among the first $(n-1)$ games, player $A$ must have won exactly $4$ games, and player $B$ must have won $(n-5)$ games.

The number of ways for player $A$ to win in $n$ games is given by selecting $4$ games for player $A$ to win out of the first $(n-1)$ games, which is $\dbinom{n-1}{4}$.

We consider all possible values of $n$ (the total number of games played):

1. Series ends in 5 games ($n = 5$): Player $A$ wins all 5 games. $\text{Ways} = \binom{4}{4} = 1$

2. Series ends in 6 games ($n = 6$): Player $A$ wins 4 out of the first 5 games and wins the $6^{\text{th}}$ game. $\text{Ways} = \binom{5}{4} = 5$

3. Series ends in 7 games ($n = 7$): Player $A$ wins 4 out of the first 6 games and wins the $7^{\text{th}}$ game. $\text{Ways} = \binom{6}{4} = 15$

4. Series ends in 8 games ($n = 8$): Player $A$ wins 4 out of the first 7 games and wins the $8^{\text{th}}$ game. $\text{Ways} = \binom{7}{4} = 35$

5. Series ends in 9 games ($n = 9$): Player $A$ wins 4 out of the first 8 games and wins the $9^{\text{th}}$ game. $\text{Ways} = \binom{8}{4} = 70$

To find the total number of ways for player $A$ to win the series, we sum the ways across all possible total number of games:

$\text{Total Ways} = \binom{4}{4} + \binom{5}{4} + \binom{6}{4} + \binom{7}{4} + \binom{8}{4}$

Using the Hockey-Stick Identity $\sum_{i=k}^{n} \binom{i}{k} = \binom{n+1}{k+1}$:

$\text{Total Ways} = \binom{8+1}{4+1} = \binom{9}{5}$

Calculating $\binom{9}{5}$:

$\binom{9}{5} = \frac{9 \times 8 \times 7 \times 6 \times 5}{5 \times 4 \times 3 \times 2 \times 1} = 126$

Thus, the number of ways in which player A wins the series is 126.