The coefficient of `x^2012` in the expansion of `(1 - x)^2008 (1+x+x^2)^2007`, is

Options

A. `""^(2012)C_(2007)`
B. `""^(2012)C_(2008)`
C. `""^(2012)C_(2009)`
D. none of these

Explanation

To find the coefficient of \( x^{2012} \) in the expansion of \( (1 - x)^{2008} (1 + x + x^2)^{2007} \), we can follow these steps:

### Step 1: Simplify the Expression
We know that \( (1 + x + x^2) \) can be rewritten using the identity:
\[ 1 + x + x^2 = \frac{1 - x^3}{1 - x} \]
Thus, we can express \( (1 + x + x^2)^{2007} \) as:
\[ (1 + x + x^2)^{2007} = \left( \frac{1 - x^3}{1 - x} \right)^{2007} = (1 - x^3)^{2007} (1 - x)^{-2007} \]

### Step 2: Substitute Back into the Original Expression
Now substituting back into the original expression, we have:
\[ (1 - x)^{2008} (1 + x + x^2)^{2007} = (1 - x)^{2008} (1 - x^3)^{2007} (1 - x)^{-2007} \]
This simplifies to:
\[ (1 - x)^{2008 - 2007} (1 - x^3)^{2007} = (1 - x)^{1} (1 - x^3)^{2007} \]
Thus, we have:
\[ (1 - x)(1 - x^3)^{2007} \]

### Step 3: Expand the Expression
Next, we need to expand \( (1 - x)(1 - x^3)^{2007} \). The term \( (1 - x^3)^{2007} \) can be expanded using the binomial theorem:
\[ (1 - x^3)^{2007} = \sum_{k=0}^{2007} \binom{2007}{k} (-1)^k x^{3k} \]
Thus, the expansion of \( (1 - x)(1 - x^3)^{2007} \) becomes:
\[ (1 - x) \sum_{k=0}^{2007} \binom{2007}{k} (-1)^k x^{3k} = \sum_{k=0}^{2007} \binom{2007}{k} (-1)^k x^{3k} - \sum_{k=0}^{2007} \binom{2007}{k} (-1)^k x^{3k + 1} \]

### Step 4: Identify the Coefficient of \( x^{2012} \)
We need to find the coefficient of \( x^{2012} \) in the above expression.

1. From the first sum \( \sum_{k=0}^{2007} \binom{2007}{k} (-1)^k x^{3k} \), we need \( 3k = 2012 \). This gives \( k = \frac{2012}{3} \), which is not an integer.

2. From the second sum \( -\sum_{k=0}^{2007} \binom{2007}{k} (-1)^k x^{3k + 1} \), we need \( 3k + 1 = 2012 \) which gives \( 3k = 2011 \) or \( k = \frac{2011}{3} \), which is also not an integer.

Since both cases do not yield an integer \( k \), we conclude that there is no term \( x^{2012} \) in the expansion.

### Final Answer
Thus, the coefficient of \( x^{2012} \) in the expansion of \( (1 - x)^{2008} (1 + x + x^2)^{2007} \) is:
\[ \boxed{0} \]