The coefficient of `x^(18)` in the product `(1+x)(1-x)^(10)(1+x+x^(2))^(9)` is

Options

A. 84
B. `-126`
C. `-84`
D. `126`

Answer

To find the coefficient of \( x^{18} \) in the product \( (1+x)(1-x)^{10}(1+x+x^2)^{9} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1+x)(1-x)^{10}(1+x+x^2)^{9} \]
We can rewrite \( (1+x+x^2)^{9} \) using the identity for \( 1+x+x^2 \):
\[ 1+x+x^2 = \frac{1-x^3}{1-x} \]
Thus,
\[ (1+x+x^2)^{9} = \left( \frac{1-x^3}{1-x} \right)^{9} = (1-x^3)^{9}(1-x)^{-9} \]

### Step 2: Substitute back into the expression
Now substituting this back into our expression gives:
\[ (1+x)(1-x)^{10}(1-x^3)^{9}(1-x)^{-9} \]
This simplifies to:
\[ (1+x)(1-x)^{1}(1-x^3)^{9} \]
since \( (1-x)^{10} \cdot (1-x)^{-9} = (1-x)^{1} \).

### Step 3: Expand the expression
Now we need to expand:
\[ (1+x)(1-x)(1-x^3)^{9} \]
First, we expand \( (1+x)(1-x) \):
\[ (1+x)(1-x) = 1 - x^2 \]
Thus, our expression becomes:
\[ (1 - x^2)(1-x^3)^{9} \]

### Step 4: Expand \( (1-x^3)^{9} \)
Using the Binomial Theorem, we expand \( (1-x^3)^{9} \):
\[ (1-x^3)^{9} = \sum_{k=0}^{9} \binom{9}{k} (-1)^k x^{3k} \]

### Step 5: Combine the expansions
Now we need to find the coefficient of \( x^{18} \) in:
\[ (1 - x^2) \sum_{k=0}^{9} \binom{9}{k} (-1)^k x^{3k} \]
This can be split into two parts:
1. The term from \( 1 \cdot \sum_{k=0}^{9} \binom{9}{k} (-1)^k x^{3k} \)
2. The term from \( -x^2 \cdot \sum_{k=0}^{9} \binom{9}{k} (-1)^k x^{3k} \)

### Step 6: Find the coefficient of \( x^{18} \)
1. From \( 1 \cdot \sum_{k=0}^{9} \binom{9}{k} (-1)^k x^{3k} \), we need \( 3k = 18 \) which gives \( k = 6 \). The coefficient is:
\[ \binom{9}{6} (-1)^6 = \binom{9}{6} = 84 \]

2. From \( -x^2 \cdot \sum_{k=0}^{9} \binom{9}{k} (-1)^k x^{3k} \), we need \( 3k + 2 = 18 \) which gives \( 3k = 16 \) or \( k = \frac{16}{3} \), which is not an integer, so this contributes \( 0 \).

### Final Step: Combine the results
Thus, the total coefficient of \( x^{18} \) is:
\[ 84 + 0 = 84 \]

### Conclusion
The coefficient of \( x^{18} \) in the product \( (1+x)(1-x)^{10}(1+x+x^2)^{9} \) is \( \boxed{84} \).