In a unique hockey series between India & Pakistan, they decide to play 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 of these

Answer

To find 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 problem as follows:

### Step-by-Step Solution:

1. **Understanding the Winning Condition**:
India must win exactly 5 matches to win the series. The series can end with India winning the 5th match after winning a certain number of matches and losing some matches.

2. **Possible Match Outcomes**:
Let 'W' represent a win for India and 'L' represent a loss for India. The series can end in the following scenarios before India wins the 5th match:
- 4 wins and 0 losses (WWWWW)
- 4 wins and 1 loss (WWWLW)
- 4 wins and 2 losses (WWWLWL)
- 4 wins and 3 losses (WWLWLW)
- 4 wins and 4 losses (WLWLWL)

3. **Counting the Ways**:
For each scenario, we will calculate the number of ways India can arrange its wins and losses before the final win (the 5th win).

- **Case 1**: 4 wins and 0 losses
- The only sequence is WWWWW.
- Number of ways = 1.

- **Case 2**: 4 wins and 1 loss
- We can arrange 4 wins and 1 loss in the first 5 matches, followed by a win.
- The number of arrangements = \( \binom{5}{1} = 5 \).

- **Case 3**: 4 wins and 2 losses
- We can arrange 4 wins and 2 losses in the first 6 matches, followed by a win.
- The number of arrangements = \( \binom{6}{2} = 15 \).

- **Case 4**: 4 wins and 3 losses
- We can arrange 4 wins and 3 losses in the first 7 matches, followed by a win.
- The number of arrangements = \( \binom{7}{3} = 35 \).

- **Case 5**: 4 wins and 4 losses
- We can arrange 4 wins and 4 losses in the first 8 matches, followed by a win.
- The number of arrangements = \( \binom{8}{4} = 70 \).

4. **Total Number of Ways**:
Now, we sum up all the ways from each case:
\[ \text{Total Ways} = 1 + 5 + 15 + 35 + 70 = 126 \]

Thus, the total number of ways in which India can win the series is **126**.