The coefficient of `x^(10)` in the expansion of `(1+x)^(2)(1+x^(2))^(3)(1+x^(3))^(4)` is equal to

Options

A. 50
B. 52
C. 44
D. 56

Answer

To find the coefficient of \( x^{10} \) in the expansion of \( (1+x)^2 (1+x^2)^3 (1+x^3)^4 \), we can break down the problem into manageable parts.

### Step-by-Step Solution:

1. **Expand Each Factor**:
- The first factor is \( (1+x)^2 \).
\[ (1+x)^2 = 1 + 2x + x^2 \]
- The second factor is \( (1+x^2)^3 \).
\[ (1+x^2)^3 = 1 + 3x^2 + 3x^4 + x^6 \]
- The third factor is \( (1+x^3)^4 \).
\[ (1+x^3)^4 = 1 + 4x^3 + 6x^6 + 4x^9 + x^{12} \]

2. **Combine the Expansions**:
We need to find the coefficient of \( x^{10} \) in the product of these three expansions:
\[ (1 + 2x + x^2)(1 + 3x^2 + 3x^4 + x^6)(1 + 4x^3 + 6x^6 + 4x^9 + x^{12}) \]

3. **Identify Combinations that Yield \( x^{10} \)**:
We can find combinations of terms from each expansion that sum to \( x^{10} \):
- From \( (1+x)^2 \): Choose \( 1, 2x, x^2 \)
- From \( (1+x^2)^3 \): Choose \( 1, 3x^2, 3x^4, x^6 \)
- From \( (1+x^3)^4 \): Choose \( 1, 4x^3, 6x^6, 4x^9, x^{12} \)

We need to find combinations of these terms that add up to \( x^{10} \):
- **Combination 1**: \( 1 \) from \( (1+x)^2 \), \( 3x^4 \) from \( (1+x^2)^3 \), and \( 4x^6 \) from \( (1+x^3)^4 \):
\[ 1 \cdot 3 \cdot 4 = 12 \]
- **Combination 2**: \( 2x \) from \( (1+x)^2 \), \( 3x^4 \) from \( (1+x^2)^3 \), and \( 4x^9 \) from \( (1+x^3)^4 \):
\[ 2 \cdot 3 \cdot 4 = 24 \]
- **Combination 3**: \( x^2 \) from \( (1+x)^2 \), \( 3x^4 \) from \( (1+x^2)^3 \), and \( 4x^9 \) from \( (1+x^3)^4 \):
\[ 1 \cdot 3 \cdot 4 = 12 \]
- **Combination 4**: \( 2x \) from \( (1+x)^2 \), \( 3x^2 \) from \( (1+x^2)^3 \), and \( 4x^6 \) from \( (1+x^3)^4 \):
\[ 2 \cdot 3 \cdot 4 = 24 \]

4. **Sum the Coefficients**:
Now we sum all the contributions:
\[ 12 + 24 + 12 + 24 = 72 \]

5. **Final Result**:
The coefficient of \( x^{10} \) in the expansion of \( (1+x)^2 (1+x^2)^3 (1+x^3)^4 \) is \( 72 \).