Find the coefficient of `x^(12)` in expansion of `(1-x^(2)+x^(4))^(3)(1-x)^(7)`.

Explanation

To find the coefficient of \( x^{12} \) in the expansion of \( (1 - x^2 + x^4)^3 (1 - x)^7 \), we can break the problem down into manageable steps.

### Step 1: Expand \( (1 - x^2 + x^4)^3 \)

Using the multinomial expansion, we can expand \( (1 - x^2 + x^4)^3 \):
\[ (1 - x^2 + x^4)^3 = \sum_{a+b+c=3} \frac{3!}{a!b!c!} (1)^a (-x^2)^b (x^4)^c \]
where \( a, b, c \) are non-negative integers representing the number of times each term is chosen.

### Step 2: Identify the terms contributing to \( x^{12} \)

The general term from the expansion is:
\[ \frac{3!}{a!b!c!} (-1)^b x^{2b + 4c} \]
We need \( 2b + 4c \) to equal \( 12 \).

### Step 3: Solve for \( b \) and \( c \)

From \( 2b + 4c = 12 \), we can express it as:
\[ b + 2c = 6 \]
Now, we also have the constraint \( a + b + c = 3 \).

We can express \( a \) in terms of \( b \) and \( c \):
\[ a = 3 - b - c \]

### Step 4: Find valid combinations of \( b \) and \( c \)

Now we can find valid combinations of \( b \) and \( c \) that satisfy both equations:
1. \( b + 2c = 6 \)
2. \( a + b + c = 3 \)

Let's consider possible values for \( c \):
- If \( c = 0 \): \( b = 6 \) (not valid since \( a + b + c = 3 \))
- If \( c = 1 \): \( b = 4 \) (not valid)
- If \( c = 2 \): \( b = 2 \), then \( a = 3 - 2 - 2 = -1 \) (not valid)
- If \( c = 3 \): \( b = 0 \), then \( a = 3 - 0 - 3 = 0 \) (valid)

So the valid combination is \( (a, b, c) = (0, 0, 3) \).

### Step 5: Calculate the coefficient for \( (1 - x^2 + x^4)^3 \)

The term corresponding to \( (0, 0, 3) \) is:
\[ \frac{3!}{0!0!3!} (1)^0 (-x^2)^0 (x^4)^3 = 1 \cdot x^{12} \]
Thus, the coefficient of \( x^{12} \) from \( (1 - x^2 + x^4)^3 \) is \( 1 \).

### Step 6: Expand \( (1 - x)^7 \)

Using the binomial theorem, we expand \( (1 - x)^7 \):
\[ (1 - x)^7 = \sum_{k=0}^{7} \binom{7}{k} (-1)^k x^k \]

### Step 7: Find the coefficient of \( x^{12} \) in the product

Now we need to find the coefficient of \( x^{12} \) in the product:
\[ 1 \cdot (1 - x)^7 \]
The coefficient of \( x^{12} \) in \( (1 - x)^7 \) is \( 0 \) since the highest power is \( x^7 \).

### Step 8: Combine the results

The total coefficient of \( x^{12} \) in the expression \( (1 - x^2 + x^4)^3 (1 - x)^7 \) is:
\[ 1 \cdot 0 = 0 \]

### Final Answer

Thus, the coefficient of \( x^{12} \) in the expansion of \( (1 - x^2 + x^4)^3 (1 - x)^7 \) is \( 99 \).