To find the coefficient of \( x^{2009} \) in the expression \( (1 + x + x^2 + x^3 + x^4)^{1001} (1 - x)^{1002} \), we can follow these steps:
### Step 1: Simplify the first part of the expression
The expression \( 1 + x + x^2 + x^3 + x^4 \) can be recognized 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 there are 5 terms (from \( x^0 \) to \( x^4 \)). Thus, we can write:
\[
1 + x + x^2 + x^3 + x^4 = \frac{1 - x^5}{1 - x}
\]
### Step 2: Substitute back into the expression
Now we substitute this back into our original expression:
\[
(1 + x + x^2 + x^3 + x^4)^{1001} = \left( \frac{1 - x^5}{1 - x} \right)^{1001}
\]
This gives us:
\[
\frac{(1 - x^5)^{1001}}{(1 - x)^{1001}}
\]
So the entire expression becomes:
\[
\frac{(1 - x^5)^{1001} (1 - x)^{-1002}}{(1 - x)^{1001}} = (1 - x^5)^{1001} (1 - x)^{-1002}
\]
### Step 3: Expand using the Binomial Theorem
Next, we need to expand both parts using the Binomial Theorem.
1. **For \( (1 - x^5)^{1001} \)**:
The expansion will give us terms of the form \( \binom{1001}{k} (-1)^k x^{5k} \).
2. **For \( (1 - x)^{-1002} \)**:
The expansion will give us terms of the form \( \binom{n+k-1}{k} x^k \) where \( n = 1002 \).
### Step 4: Find the coefficient of \( x^{2009} \)
We need to find the coefficient of \( x^{2009} \) in the product of these two expansions.
From \( (1 - x^5)^{1001} \), we have \( x^{5k} \) and from \( (1 - x)^{-1002} \), we have \( x^m \). We need:
\[
5k + m = 2009
\]
This can be rearranged to:
\[
m = 2009 - 5k
\]
### Step 5: Determine valid values for \( k \)
Since \( m \) must be non-negative, we have:
\[
2009 - 5k \geq 0 \implies k \leq \frac{2009}{5} = 401.8
\]
Thus, \( k \) can take integer values from 0 to 401.
### Step 6: Check if \( 2009 - 5k \) is valid
Now, we need to check if \( 2009 - 5k \) is a non-negative integer for each \( k \).
1. If \( k = 0 \), \( m = 2009 \)
2. If \( k = 1 \), \( m = 2004 \)
3. If \( k = 2 \), \( m = 1999 \)
4. Continuing this way, we notice that \( 2009 - 5k \) will never yield an integer for \( k \) values that would make \( m \) equal to an odd number.
### Conclusion
Since \( 2009 \) is not expressible as \( 5k + m \) where both \( k \) and \( m \) are non-negative integers, the coefficient of \( x^{2009} \) in the expansion is:
\[
\text{Coefficient of } x^{2009} = 0
\]
Thus, the answer is (a) 0.
