Two players $$A$$ and $$B$$ play a series of badminton games. The first player, who wins 5 games  first, wins the series. Assuming that no game ends in a draw, the number of ways in which player $$A$$ wins the series is :_____.

To find the number of ways player $$A$$ can win the series, we need to consider all the different scenarios based on the total number of games played.

Since the first player to win 5 games wins the series, the series can last anywhere from 5 games to a maximum of 9 games (because if they played 10 games, one player would have already reached 5 wins).

Crucial Rule

For player $$A$$ to win in exactly $$n$$ games, player $$A$$ must win the very last game (the $$n^{th}$$ game). This means in the preceding $$n-1$$ games, player $$A$$ must have won exactly 4 games.

Step-by-Step Calculation

We sum the number of ways for $$A$$ to win for each possible series length:

Total Number of Ways

Now, we add these individual scenarios together:

$$\text{Total} = 1 + 5 + 15 + 35 + 70$$

$$\text{Total} = 126$$

The number of ways in which player $$A$$ wins the series is 126.