The coefficient of `x^(4)` in the expansion `(1-x)^(8)(1+x)^(12)` is equal to

Options

A. 1
B. `-48`
C. 28
D. `-19`

Answer

To find the coefficient of \( x^4 \) 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 desired coefficient.

### Step 1: Expand \( (1-x)^8 \) using the binomial theorem
The binomial expansion of \( (1-x)^n \) is given by:
\[ (1-x)^n = \sum_{k=0}^{n} \binom{n}{k} (-x)^k \]
For \( n = 8 \):
\[ (1-x)^8 = \sum_{k=0}^{8} \binom{8}{k} (-1)^k x^k \]
This gives us:
\[ 1 - 8x + 28x^2 - 56x^3 + 70x^4 - 56x^5 + 28x^6 - 8x^7 + x^8 \]

### Step 2: Expand \( (1+x)^{12} \) using the binomial theorem
The binomial expansion of \( (1+x)^n \) is given by:
\[ (1+x)^n = \sum_{j=0}^{n} \binom{n}{j} x^j \]
For \( n = 12 \):
\[ (1+x)^{12} = \sum_{j=0}^{12} \binom{12}{j} x^j \]
This gives us:
\[ 1 + 12x + 66x^2 + 220x^3 + 495x^4 + 792x^5 + 924x^6 + 792x^7 + 495x^8 + 220x^9 + 66x^{10} + 12x^{11} + x^{12} \]

### Step 3: Combine the expansions to find the coefficient of \( x^4 \)
We need to find the coefficient of \( x^4 \) in the product \( (1-x)^8(1+x)^{12} \). This can be done by considering the contributions from different combinations of terms from each expansion that multiply to give \( x^4 \).

The relevant combinations are:
1. \( \binom{8}{0}(-1)^0 \cdot \binom{12}{4} = 1 \cdot 495 \)
2. \( \binom{8}{1}(-1)^1 \cdot \binom{12}{3} = -8 \cdot 220 \)
3. \( \binom{8}{2}(-1)^2 \cdot \binom{12}{2} = 28 \cdot 66 \)
4. \( \binom{8}{3}(-1)^3 \cdot \binom{12}{1} = -56 \cdot 12 \)
5. \( \binom{8}{4}(-1)^4 \cdot \binom{12}{0} = 70 \cdot 1 \)

Now we compute each term:
1. \( 1 \cdot 495 = 495 \)
2. \( -8 \cdot 220 = -1760 \)
3. \( 28 \cdot 66 = 1848 \)
4. \( -56 \cdot 12 = -672 \)
5. \( 70 \cdot 1 = 70 \)

### Step 4: Add the contributions
Now, we sum these contributions:
\[ 495 - 1760 + 1848 - 672 + 70 \]
Calculating step by step:
- \( 495 - 1760 = -1265 \)
- \( -1265 + 1848 = 583 \)
- \( 583 - 672 = -89 \)
- \( -89 + 70 = -19 \)

### Final Result
Thus, the coefficient of \( x^4 \) in the expansion of \( (1-x)^8(1+x)^{12} \) is \( -19 \).