Question

Two players A and B play a series of games of badminton. The 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 _________.

Answer 1

Correct Answer: 126

Step 1: Understanding the Question:
We need to count the number of possible sequences of game outcomes where player A is the first to win 5 games. The total number of games played is not fixed.
Step 2: Key Formula or Approach:
For player A to win the series, A must win the last game played. The series can last for 5, 6, 7, 8, or 9 games (if A wins, B can have at most 4 wins). We will consider each case for the total number of games and sum the results. If A wins in $n$ games, A must have won exactly 4 of the first $n-1$ games, and then won the $n$-th game. The number of ways for this is $\binom{n-1}{4}$.
Step 3: Detailed Explanation:
Let's analyze the number of ways A can win based on the total number of games played.
- A wins in 5 games: A must win all 5 games. The sequence is AAAAA. The 5th game must be a win for A. The previous 4 games must also be wins for A. The number of ways is $\binom{4}{4} = 1$.
- A wins in 6 games: A must win the 6th game. In the first 5 games, A must have won 4 times and B must have won once. The number of ways to arrange this is $\binom{5}{4} = 5$.
- A wins in 7 games: A must win the 7th game. In the first 6 games, A must have won 4 times. The number of ways is $\binom{6}{4} = \frac{6 \times 5}{2} = 15$.
- A wins in 8 games: A must win the 8th game. In the first 7 games, A must have won 4 times. The number of ways is $\binom{7}{4} = \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$.
- A wins in 9 games: A must win the 9th game. In the first 8 games, A must have won 4 times. The number of ways is $\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$.
The total number of ways for A to win the series is the sum of all these cases:
Total ways = $1 + 5 + 15 + 35 + 70 = 126$.
Step 4: Final Answer:
The total number of ways player A wins the series is 126.