The coefficients of `x^(6)` in ` (1 + x + x^(2))^(-3)`, is

Options

A. 2
B. 3
C. 4
D. 6

Explanation

To find the coefficient of \( x^6 \) in \( (1 + x + x^2)^{-3} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \( (1 + x + x^2)^{-3} \). We can rewrite this using the identity for negative exponents and the binomial theorem.

\[ (1 + x + x^2)^{-3} = \frac{1}{(1 + x + x^2)^3} \]

### Step 2: Simplify the expression
Next, we can simplify \( 1 + x + x^2 \) by multiplying the numerator and denominator by \( 1 - x \):

\[ 1 + x + x^2 = \frac{(1 - x)(1 + x + x^2)}{1 - x} = \frac{1 - x^3}{1 - x} \]

Thus, we have:

\[ (1 + x + x^2)^{-3} = \left( \frac{1 - x^3}{1 - x} \right)^{-3} = \frac{(1 - x)^{3}}{(1 - x^3)^{-3}} \]

### Step 3: Expand using the Binomial Theorem
Now we can express this as:

\[ (1 - x)^3 \cdot (1 - x^3)^{3} \]

Using the binomial theorem, we can expand both expressions:

1. **For \( (1 - x)^3 \)**:
\[ (1 - x)^3 = 1 - 3x + 3x^2 - x^3 \]

2. **For \( (1 - x^3)^{-3} \)**:
Using the binomial theorem for negative exponents:
\[ (1 - x^3)^{-3} = \sum_{n=0}^{\infty} \binom{n+2}{2} x^{3n} \]

### Step 4: Find the coefficient of \( x^6 \)
To find the coefficient of \( x^6 \), we need to consider the products of terms from both expansions that yield \( x^6 \).

From \( (1 - 3x + 3x^2 - x^3) \):
- The term \( 1 \) from \( (1 - x)^3 \) will multiply with the coefficient of \( x^6 \) in \( (1 - x^3)^{-3} \), which is \( \binom{6/3 + 2}{2} = \binom{4}{2} = 6 \).

- The term \( -3x \) from \( (1 - x)^3 \) will multiply with the coefficient of \( x^5 \) in \( (1 - x^3)^{-3} \), which is \( \binom{5/3 + 2}{2} = 0 \) (as \( 5/3 \) is not an integer).

- The term \( 3x^2 \) from \( (1 - x)^3 \) will multiply with the coefficient of \( x^4 \) in \( (1 - x^3)^{-3} \), which is \( \binom{4/3 + 2}{2} = 0 \) (as \( 4/3 \) is not an integer).

- The term \( -x^3 \) from \( (1 - x)^3 \) will multiply with the coefficient of \( x^3 \) in \( (1 - x^3)^{-3} \), which is \( \binom{3/3 + 2}{2} = \binom{3}{2} = 3 \).

### Step 5: Combine the coefficients
Now we can combine the contributions:

\[ \text{Coefficient of } x^6 = 1 \cdot 6 + (-3) \cdot 0 + 3 \cdot 0 + (-1) \cdot 3 = 6 - 3 = 3 \]

### Final Answer
The coefficient of \( x^6 \) in \( (1 + x + x^2)^{-3} \) is \( \boxed{3} \).