Find the number of terms in the expansion of `(1+3x+3x^(2)+x^(3))^(15)`

Explanation

To find the number of terms in the expansion of \((1 + 3x + 3x^2 + x^3)^{15}\), we can follow these steps:

### Step 1: Simplify the expression
First, we recognize that the expression \(1 + 3x + 3x^2 + x^3\) can be rewritten. We notice that this is a polynomial that can be expressed as:

\[ 1 + 3x + 3x^2 + x^3 = (1 + x)^3 \]

This is because the binomial expansion of \((1 + x)^3\) gives us:

\[ 1 + 3x + 3x^2 + x^3 \]

### Step 2: Rewrite the original expression
Now we can rewrite the original expression as:

\[ (1 + 3x + 3x^2 + x^3)^{15} = ((1 + x)^3)^{15} \]

### Step 3: Simplify further
This simplifies to:

\[ (1 + x)^{45} \]

### Step 4: Determine the number of terms
To find the number of terms in the expansion of \((1 + x)^{45}\), we can use the formula for the number of terms in the binomial expansion, which is given by \(n + 1\), where \(n\) is the exponent.

Here, \(n = 45\), so the number of terms is:

\[ 45 + 1 = 46 \]

### Final Answer
Thus, the number of terms in the expansion of \((1 + 3x + 3x^2 + x^3)^{15}\) is **46**.

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