The coefficient of `x^(7)` in the expansion of `(1-x-x^(3)+x^(4))^(8)` is equal to

Options

A. (a) `-648`
B. (b) `792`
C. (c) `-792`
D. (d) `648`

Explanation

To find the coefficient of \( x^7 \) in the expansion of \( (1 - x - x^3 + x^4)^8 \), we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression:
\[ (1 - x - x^3 + x^4)^8 \]
We can factor this expression. Notice that we can group the terms:
\[ = (1 - x + x^4 - x^3)^8 \]
This can be rewritten as:
\[ = (1 - x + x^4 - x^3)^8 = (1 - x(1 + x^2))^8 \]

### Step 2: Apply the Binomial Theorem
Using the binomial theorem, we can expand \( (1 - x(1 + x^2))^8 \):
\[ = \sum_{k=0}^{8} \binom{8}{k} (-x(1 + x^2))^k \]
This simplifies to:
\[ = \sum_{k=0}^{8} \binom{8}{k} (-1)^k x^k (1 + x^2)^k \]

### Step 3: Expand \( (1 + x^2)^k \)
Now, we need to expand \( (1 + x^2)^k \):
\[ (1 + x^2)^k = \sum_{j=0}^{k} \binom{k}{j} x^{2j} \]
Combining this with our previous expansion, we have:
\[ = \sum_{k=0}^{8} \binom{8}{k} (-1)^k x^k \sum_{j=0}^{k} \binom{k}{j} x^{2j} \]

### Step 4: Find the Coefficient of \( x^7 \)
We need to find the coefficient of \( x^7 \). This can occur in two ways:
1. \( k + 2j = 7 \)
2. \( k \) must be odd since \( 7 - k \) must be even.

We can derive the possible values of \( k \) and \( j \):
- If \( k = 1 \), then \( 1 + 2j = 7 \) gives \( j = 3 \).
- If \( k = 3 \), then \( 3 + 2j = 7 \) gives \( j = 2 \).
- If \( k = 5 \), then \( 5 + 2j = 7 \) gives \( j = 1 \).
- If \( k = 7 \), then \( 7 + 2j = 7 \) gives \( j = 0 \).

### Step 5: Calculate Each Contribution
1. For \( k = 1, j = 3 \):
\[ \text{Contribution} = \binom{8}{1} (-1)^1 \binom{1}{3} = 8 \cdot (-1) \cdot 0 = 0 \]
2. For \( k = 3, j = 2 \):
\[ \text{Contribution} = \binom{8}{3} (-1)^3 \binom{3}{2} = 56 \cdot (-1) \cdot 3 = -168 \]
3. For \( k = 5, j = 1 \):
\[ \text{Contribution} = \binom{8}{5} (-1)^5 \binom{5}{1} = 56 \cdot (-1) \cdot 5 = -280 \]
4. For \( k = 7, j = 0 \):
\[ \text{Contribution} = \binom{8}{7} (-1)^7 \binom{7}{0} = 8 \cdot (-1) \cdot 1 = -8 \]

### Step 6: Sum the Contributions
Now, we sum all contributions:
\[ 0 - 168 - 280 - 8 = -456 \]

### Final Answer
Thus, the coefficient of \( x^7 \) in the expansion of \( (1 - x - x^3 + x^4)^8 \) is:
\[ \boxed{-456} \]