The coefficient of `x^(48)` in the expansion of `(1+x^(4))(1+x^(24))(1+x^(48))` is

Options

A. `""^(12)C_(6)+3`
B. `""^(12)C_(6)`
C. `1`
D. `""^(12)C_(6)+2`

Answer

To find the coefficient of \( x^{48} \) in the expansion of \( (1 + x^4)(1 + x^{24})(1 + x^{48}) \), we can follow these steps:

### Step 1: Understand the Expansion
The expression \( (1 + x^4)(1 + x^{24})(1 + x^{48}) \) can be expanded by considering all possible products of the terms from each factor.

### Step 2: Identify the Terms
Each factor contributes either 1 or \( x^k \) where \( k \) is the exponent in that factor. Therefore, we can choose:
- From \( (1 + x^4) \): either \( 1 \) or \( x^4 \)
- From \( (1 + x^{24}) \): either \( 1 \) or \( x^{24} \)
- From \( (1 + x^{48}) \): either \( 1 \) or \( x^{48} \)

### Step 3: List Possible Combinations
To get \( x^{48} \), we can have the following combinations:
1. Choose \( x^{48} \) from \( (1 + x^{48}) \) and \( 1 \) from the other two factors.
2. Choose \( x^{24} \) from \( (1 + x^{24}) \) and \( x^{24} \) from \( (1 + x^4) \) (but this is not possible since \( 4 + 24 = 28 \)).
3. Choose \( x^4 \) from \( (1 + x^4) \) and \( x^{24} \) from \( (1 + x^{24}) \) (but this gives \( 4 + 24 = 28 \)).
4. Choose \( 1 \) from all factors (which gives \( x^0 \)).

The only valid combination that gives \( x^{48} \) is:
- \( 1 \) from \( (1 + x^4) \)
- \( 1 \) from \( (1 + x^{24}) \)
- \( x^{48} \) from \( (1 + x^{48}) \)

### Step 4: Count the Coefficients
From the valid combination, we see that there is only one way to achieve \( x^{48} \):
- \( x^{48} \) contributes a coefficient of \( 1 \).

### Conclusion
Thus, the coefficient of \( x^{48} \) in the expansion of \( (1 + x^4)(1 + x^{24})(1 + x^{48}) \) is \( 1 \).

### Final Answer
The coefficient of \( x^{48} \) is \( \boxed{1} \).