Coefficient of `x^(2009)` in `(1+x+x^(2)+x^(3)+x^(4))^(1001) (1-x)^(1002)` is

Options

A. `0`
B. `4."^(1001)C_(501)`
C. `-2009`
D. none of these

Explanation

To find the coefficient of \( x^{2009} \) in the expression \( (1 + x + x^2 + x^3 + x^4)^{1001} (1 - x)^{1002} \), we will follow these steps:

### Step 1: Simplify the first part of the expression
The term \( 1 + x + x^2 + x^3 + x^4 \) can be expressed as a geometric series. The sum of a geometric series can be calculated using the formula:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a = 1 \), \( r = x \), and \( n = 5 \).
Thus, we can rewrite:
\[ 1 + x + x^2 + x^3 + x^4 = \frac{1 - x^5}{1 - x} \]
Therefore, we have:
\[ (1 + x + x^2 + x^3 + x^4)^{1001} = \left(\frac{1 - x^5}{1 - x}\right)^{1001} = (1 - x^5)^{1001} \cdot (1 - x)^{-1001} \]

### Step 2: Combine the two parts
Now, we combine this with the second part of the expression:
\[ (1 - x)^{-1001} \cdot (1 - x)^{-1002} \cdot (1 - x^5)^{1001} = (1 - x)^{-2003} \cdot (1 - x^5)^{1001} \]

### Step 3: Expand the expression
We need to find the coefficient of \( x^{2009} \) in:
\[ (1 - x)^{-2003} \cdot (1 - x^5)^{1001} \]
Using the binomial series expansion:
\[ (1 - x)^{-n} = \sum_{k=0}^{\infty} \binom{n+k-1}{k} x^k \]
we can express:
\[ (1 - x)^{-2003} = \sum_{m=0}^{\infty} \binom{2002 + m}{m} x^m \]
For \( (1 - x^5)^{1001} \), we can use the binomial theorem:
\[ (1 - x^5)^{1001} = \sum_{j=0}^{1001} \binom{1001}{j} (-1)^j x^{5j} \]

### Step 4: Find the coefficient of \( x^{2009} \)
To find the coefficient of \( x^{2009} \), we need to consider terms from both expansions that sum to \( 2009 \):
\[ m + 5j = 2009 \]
This can be rearranged to find \( m \):
\[ m = 2009 - 5j \]
Substituting this into the coefficient expressions:
\[ \text{Coefficient of } x^{2009} = \sum_{j=0}^{\lfloor 2009/5 \rfloor} \binom{2002 + (2009 - 5j)}{2009 - 5j} \cdot \binom{1001}{j} (-1)^j \]
The maximum value of \( j \) is \( \lfloor 2009/5 \rfloor = 401 \).

### Step 5: Check for valid \( j \)
We need to check if \( 2009 - 5j \) is non-negative:
- For \( j = 401 \), \( m = 2009 - 5 \times 401 = 4 \) (valid)
- For \( j = 400 \), \( m = 2009 - 5 \times 400 = 9 \) (valid)
- Continuing this way, we find that \( j \) must be such that \( 2009 - 5j \geq 0 \).

### Conclusion
However, since \( 2009 \) is not a multiple of \( 5 \), there will be no valid combinations of \( m \) and \( j \) such that \( m + 5j = 2009 \) holds true. Therefore, the coefficient of \( x^{2009} \) in the expression is:
\[ \text{Coefficient} = 0 \]

### Final Answer
The coefficient of \( x^{2009} \) is \( 0 \).