The coefficient of `x^(20)` in
` (1 + 3x + 3x^(2) + x^(3))^(20)`,I s

Options

A. `""^(60)C_(40)`
B. `""^(30)C_(20)`
C. `""^(15)C_(2)`
D. none of these

Explanation

To find the coefficient of \( x^{20} \) in the expansion of \( (1 + 3x + 3x^2 + x^3)^{20} \), we can follow these steps:

### Step 1: Simplify the expression
We notice that \( 1 + 3x + 3x^2 + x^3 \) can be rewritten using the binomial theorem. We can express it as:
\[ 1 + 3x + 3x^2 + x^3 = (1 + x^3) + 3x + 3x^2 \]
However, a more straightforward approach is to recognize that:
\[ 1 + 3x + 3x^2 + x^3 = (1 + x^3) + 3x(1 + x) \]
This doesn't simplify directly, so we will use the binomial expansion directly.

### Step 2: Rewrite the expression
We can factor out \( 1 + x^3 \) from the original expression:
\[ (1 + 3x + 3x^2 + x^3)^{20} = (1 + x^3)^{20} + \text{(other terms)} \]
However, we can also notice that:
\[ 1 + 3x + 3x^2 + x^3 = (1 + x)^3 \]
Thus, we can rewrite the expression as:
\[ (1 + x)^3 = 1 + 3x + 3x^2 + x^3 \]
So, we can express our original expression as:
\[ (1 + x)^{20} \]

### Step 3: Use the Binomial Theorem
Now, we need to find the coefficient of \( x^{20} \) in \( (1 + x)^{20} \). According to the binomial theorem:
\[ (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]
The coefficient of \( x^{20} \) in \( (1 + x)^{20} \) is given by:
\[ \binom{20}{20} = 1 \]

### Step 4: Coefficient of \( x^{20} \)
Thus, the coefficient of \( x^{20} \) in the expansion of \( (1 + 3x + 3x^2 + x^3)^{20} \) is:
\[ \binom{20}{20} = 1 \]

### Final Answer
The coefficient of \( x^{20} \) in \( (1 + 3x + 3x^2 + x^3)^{20} \) is \( 1 \).

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