Find the number of ways in which India can win the series of 11 matches (If no match is drawn and all matches are played).

Explanation

To solve the problem of finding the number of ways in which India can win a series of 11 matches (with no draws), we need to determine the number of scenarios where India wins at least 6 matches. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Winning Conditions
India must win at least 6 matches out of 11. This means we need to consider the cases where India wins 6, 7, 8, 9, 10, or all 11 matches.

### Step 2: Calculate the Number of Winning Scenarios
For each case, we can use the binomial coefficient to determine the number of ways India can win those matches. The binomial coefficient \( nCr \) gives us the number of ways to choose \( r \) successes (wins) from \( n \) trials (matches).

1. **Winning 6 matches**: The number of ways is \( \binom{11}{6} \)
2. **Winning 7 matches**: The number of ways is \( \binom{11}{7} \)
3. **Winning 8 matches**: The number of ways is \( \binom{11}{8} \)
4. **Winning 9 matches**: The number of ways is \( \binom{11}{9} \)
5. **Winning 10 matches**: The number of ways is \( \binom{11}{10} \)
6. **Winning 11 matches**: The number of ways is \( \binom{11}{11} \)

### Step 3: Sum the Winning Scenarios
Now, we need to sum all these scenarios to find the total number of ways India can win the series:

\[ \text{Total Ways} = \binom{11}{6} + \binom{11}{7} + \binom{11}{8} + \binom{11}{9} + \binom{11}{10} + \binom{11}{11} \]

### Step 4: Use the Binomial Theorem
According to the binomial theorem, we know that:

\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n \]

For \( n = 11 \):

\[ \sum_{r=0}^{11} \binom{11}{r} = 2^{11} = 2048 \]

### Step 5: Calculate the Required Sum
The sum of the binomial coefficients can be split into two equal parts due to symmetry:

\[ \sum_{r=0}^{5} \binom{11}{r} = \sum_{r=6}^{11} \binom{11}{r} \]

Thus, we can express our required sum as:

\[ \sum_{r=6}^{11} \binom{11}{r} = \frac{1}{2} \times 2^{11} = 2^{10} = 1024 \]

### Final Answer
The total number of ways in which India can win the series of 11 matches is:

\[ \boxed{1024} \]

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