The coefficient of `x^(6)` in the expansion of `(1-x)^(8)(1+x)^(12)` is equal to

Options

A. 168
B. `-8`
C. 28
D. 104

Answer

To find the coefficient of \( x^6 \) in the expansion of \( (1 - x)^8 (1 + x)^{12} \), we will use the binomial theorem to expand both terms and then combine them to find the required coefficient.

### Step-by-Step Solution:

1. **Expand \( (1 - x)^8 \)**:
Using the binomial theorem, we have:
\[ (1 - x)^8 = \sum_{k=0}^{8} \binom{8}{k} (-x)^k = \sum_{k=0}^{8} \binom{8}{k} (-1)^k x^k \]
The expansion gives us terms from \( x^0 \) to \( x^8 \).

2. **Expand \( (1 + x)^{12} \)**:
Again, using the binomial theorem:
\[ (1 + x)^{12} = \sum_{j=0}^{12} \binom{12}{j} x^j \]
This expansion gives us terms from \( x^0 \) to \( x^{12} \).

3. **Combine the Expansions**:
We need to find the coefficient of \( x^6 \) in the product:
\[ (1 - x)^8 (1 + x)^{12} \]
To find the coefficient of \( x^6 \), we consider all pairs of terms \( x^k \) from \( (1 - x)^8 \) and \( x^{6-k} \) from \( (1 + x)^{12} \) such that \( k + (6 - k) = 6 \).

The pairs are:
- \( k = 0 \) and \( j = 6 \)
- \( k = 1 \) and \( j = 5 \)
- \( k = 2 \) and \( j = 4 \)
- \( k = 3 \) and \( j = 3 \)
- \( k = 4 \) and \( j = 2 \)
- \( k = 5 \) and \( j = 1 \)
- \( k = 6 \) and \( j = 0 \)

4. **Calculate Each Pair**:
- For \( k = 0, j = 6 \):
\[ \text{Coefficient} = \binom{8}{0} \cdot \binom{12}{6} = 1 \cdot 924 = 924 \]
- For \( k = 1, j = 5 \):
\[ \text{Coefficient} = \binom{8}{1} \cdot \binom{12}{5} = 8 \cdot 792 = 6336 \]
- For \( k = 2, j = 4 \):
\[ \text{Coefficient} = \binom{8}{2} \cdot \binom{12}{4} = 28 \cdot 495 = 13860 \]
- For \( k = 3, j = 3 \):
\[ \text{Coefficient} = \binom{8}{3} \cdot \binom{12}{3} = 56 \cdot 220 = 12320 \]
- For \( k = 4, j = 2 \):
\[ \text{Coefficient} = \binom{8}{4} \cdot \binom{12}{2} = 70 \cdot 66 = 4620 \]
- For \( k = 5, j = 1 \):
\[ \text{Coefficient} = \binom{8}{5} \cdot \binom{12}{1} = 56 \cdot 12 = 672 \]
- For \( k = 6, j = 0 \):
\[ \text{Coefficient} = \binom{8}{6} \cdot \binom{12}{0} = 28 \cdot 1 = 28 \]

5. **Combine the Coefficients**:
Now we combine the coefficients, remembering that the terms from \( (1 - x)^8 \) alternate in sign:
\[ \text{Total Coefficient} = 924 - 6336 + 13860 - 12320 + 4620 - 672 + 28 \]

6. **Calculate the Final Result**:
Performing the arithmetic:
\[ 924 - 6336 + 13860 - 12320 + 4620 - 672 + 28 = 104 \]

Thus, the coefficient of \( x^6 \) in the expansion of \( (1 - x)^8 (1 + x)^{12} \) is **104**.