In a unique hockey series between India and Pakistan, they decide to play on till a team wins 5 matches. The number of ways in which the series can be won by India, if no match ends in a draw is

Options

A. 126
B. 252
C. 225
D. none

Answer

To solve the problem of determining the number of ways in which India can win a unique hockey series against Pakistan (where the series continues until one team wins 5 matches), we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- India wins the series if they win 5 matches before Pakistan does.
- The last match must always be a win for India since they need to reach 5 wins to win the series.

2. **Denoting Wins and Losses**:
- Let 'W' represent a win for India and 'L' represent a win for Pakistan.
- Since India must win the series, the sequence of matches must end with 'W'.

3. **Counting Matches**:
- To find the number of ways India can win the series, we need to consider how many matches can occur before the final match (which is a win for India).
- If India wins 5 matches, the maximum number of matches played can be 9 (4 wins for India and 4 wins for Pakistan, followed by the final win for India).

4. **Possible Scenarios**:
- India can win the series in the following scenarios:
- India wins 5 matches and loses 0 matches (WWWWW).
- India wins 5 matches and loses 1 match (e.g., WWWWL).
- India wins 5 matches and loses 2 matches (e.g., WWLWW).
- India wins 5 matches and loses 3 matches (e.g., WLWLW).
- India wins 5 matches and loses 4 matches (e.g., LLLLW).

5. **Calculating Combinations**:
- For each scenario, we need to calculate the number of ways to arrange the wins and losses:
- For the case where India wins 5 and loses 0: There is only 1 way (WWWWW).
- For the case where India wins 5 and loses 1: We need to choose 4 positions for 'W' out of the first 5 matches (the last match is fixed as 'W'). This can be calculated using combinations: \( \binom{5}{4} = 5 \).
- For the case where India wins 5 and loses 2: We choose 3 positions for 'W' out of the first 6 matches. This is \( \binom{6}{4} = 15 \).
- For the case where India wins 5 and loses 3: We choose 2 positions for 'W' out of the first 7 matches. This is \( \binom{7}{4} = 35 \).
- For the case where India wins 5 and loses 4: We choose 1 position for 'W' out of the first 8 matches. This is \( \binom{8}{4} = 70 \).

6. **Summing Up the Combinations**:
- Now, we add all the combinations together:
- Total ways = \( 1 + 5 + 15 + 35 + 70 = 126 \).

### Final Answer:
The total number of ways in which India can win the series is **126**.

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