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: Expand the expression
We start with the expression:
\[ (1 + x^4)(1 + x^{24})(1 + x^{48}) \]
We can expand this by considering all possible products of the terms from each factor.

### Step 2: Identify the terms contributing to \( x^{48} \)
We need to find combinations of terms from each factor that will yield \( x^{48} \). The possible combinations are:
1. Choosing \( 1 \) from \( (1 + x^4) \), \( (1 + x^{24}) \), and \( (1 + x^{48}) \): This gives \( 1 \).
2. Choosing \( x^4 \) from \( (1 + x^4) \), \( 1 \) from \( (1 + x^{24}) \), and \( x^{48} \) from \( (1 + x^{48}) \): This gives \( x^{4 + 24 + 48} = x^{76} \) (not contributing).
3. Choosing \( x^{24} \) from \( (1 + x^{24}) \), \( 1 \) from \( (1 + x^4) \), and \( x^{48} \) from \( (1 + x^{48}) \): This gives \( x^{24 + 48} = x^{72} \) (not contributing).
4. Choosing \( x^{48} \) from \( (1 + x^{48}) \), \( 1 \) from \( (1 + x^4) \), and \( 1 \) from \( (1 + x^{24}) \): This gives \( x^{48} \).

### Step 3: Count the contributions
From the above combinations, we see that the only way to obtain \( x^{48} \) is by selecting \( x^{48} \) from the third factor and \( 1 \) from the other two factors. Thus, there is exactly **one term** that contributes to \( x^{48} \).

### Conclusion
The coefficient of \( x^{48} \) in the expansion is therefore:
\[ \text{Coefficient of } x^{48} = 1 \]

### Final Answer
The coefficient of \( x^{48} \) in the expansion of \( (1 + x^4)(1 + x^{24})(1 + x^{48}) \) is \( 1 \).

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