Question

A and B play a tennis match which will not result in a draw. The player who wins 5 rounds first will be the winner of the match. The number of ways such that A can win the match is:

• A. 126
• B. 252
• C. 63
• D. 216

Hint:

This is a classic application of the Negative Binomial distribution logic. Alternatively, you can use the identity $\sum_{r=k}^n \binom{r-1}{k-1} = \binom{n}{k}$. Here, $\binom{9}{5} = 126$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For A to be the winner by winning 5 rounds first, the match must end in exactly $n$ rounds, where $5 \le n \le 9$. The $n^{th}$ round must be won by A, and in the previous $n-1$ rounds, A must have won exactly 4 rounds.

Step 2: Key Formula or Approach:
The number of ways for A to win in exactly $n$ rounds is given by: \[ \binom{n-1}{4} \] The total number of ways is the sum of ways for all possible values of $n$.

Step 3: Detailed Explanation:
1. If A wins in 5 rounds: $\binom{4}{4} = 1$ way. 2. If A wins in 6 rounds: $\binom{5}{4} = 5$ ways. 3. If A wins in 7 rounds: $\binom{6}{4} = 15$ ways. 4. If A wins in 8 rounds: $\binom{7}{4} = 35$ ways. 5. If A wins in 9 rounds: $\binom{8}{4} = 70$ ways. Total ways = $1 + 5 + 15 + 35 + 70 = 126$.

Step 4: Final Answer:
The total number of ways A can win the match is 126.