Find the coefficient of `x^(2).y^(7)` in the expansion of `(x+2y)^(9)`

Explanation

To find the coefficient of \( x^2 y^7 \) in the expansion of \( (x + 2y)^9 \), we can use the Binomial Theorem.

### Step-by-Step Solution:

1. **Understand the Binomial Theorem**:
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, \( a = x \), \( b = 2y \), and \( n = 9 \).

2. **Set up the expansion**:
The expansion of \( (x + 2y)^9 \) can be written as:
\[ (x + 2y)^9 = \sum_{k=0}^{9} \binom{9}{k} x^{9-k} (2y)^k \]
This can be rewritten as:
\[ = \sum_{k=0}^{9} \binom{9}{k} x^{9-k} 2^k y^k \]

3. **Identify the required term**:
We need to find the coefficient of the term \( x^2 y^7 \). This means we want \( 9 - k = 2 \) (for \( x^2 \)) and \( k = 7 \) (for \( y^7 \)).

4. **Solve for k**:
From \( 9 - k = 2 \), we find:
\[ k = 9 - 2 = 7 \]

5. **Substitute k into the binomial expansion**:
Now, substitute \( k = 7 \) into the expansion:
\[ \text{Term} = \binom{9}{7} x^{2} (2y)^{7} \]

6. **Calculate the coefficient**:
The coefficient of \( x^2 y^7 \) is:
\[ \binom{9}{7} \cdot 2^7 \]
We can simplify this:
\[ \binom{9}{7} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
Now calculate \( 2^7 \):
\[ 2^7 = 128 \]
Therefore, the coefficient is:
\[ 36 \cdot 128 \]

7. **Final Calculation**:
Now, calculate \( 36 \cdot 128 \):
\[ 36 \cdot 128 = 4608 \]

### Conclusion:
The coefficient of \( x^2 y^7 \) in the expansion of \( (x + 2y)^9 \) is \( 4608 \).