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
We start with the expression:
\[ 1 + 3x + 3x^2 + x^3 \]
Notice that this expression can be rewritten as:
\[ 1 + (3x + 3x^2 + x^3) = 1 + x^3 + 3x(1 + x) \]
However, a more straightforward way to simplify is to recognize that it can be factored as:
\[ (1 + x^3) + 3x(1 + x) \]
But for our purpose, we can directly consider it as:
\[ (1 + x^3)^{15} \]

### Step 2: Recognize the Structure
The expression can be recognized as:
\[ (1 + x^3)^{15} \]
This means we can treat \(1 + x^3\) as a binomial.

### Step 3: Apply the Binomial Theorem
According to the binomial theorem, the expansion of \((a + b)^n\) contains \(n + 1\) terms. In our case, since we have:
\[ (1 + x^3)^{15} \]
the number of terms in this expansion will be:
\[ 15 + 1 = 16 \]

### Step 4: Count the Distinct Terms
Now, we need to consider the distinct terms generated when we expand \((1 + x^3)^{15}\). The terms will be of the form:
\[ x^{3k} \quad \text{for } k = 0, 1, 2, \ldots, 15 \]
This means the powers of \(x\) will be \(0, 3, 6, 9, 12, 15\).

### Step 5: Determine the Number of Distinct Terms
The powers of \(x\) are:
- \(0\) (from \(k=0\))
- \(3\) (from \(k=1\))
- \(6\) (from \(k=2\))
- \(9\) (from \(k=3\))
- \(12\) (from \(k=4\))
- \(15\) (from \(k=5\))

Thus, we have the following distinct terms:
\[ x^0, x^3, x^6, x^9, x^{12}, x^{15} \]
This gives us a total of \(6\) distinct terms.

### Final Answer
Therefore, the number of terms in the expansion of \((1 + 3x + 3x^2 + x^3)^{15}\) is:
\[ \boxed{6} \]