No. of terms in the expansion of `(1+3x+3x^(2)+x^(3))^(10)` is:

Options

A. 31
B. 32
C. 10
D. 11

Explanation

To find the number of terms in the expansion of \((1 + 3x + 3x^2 + x^3)^{10}\), we can follow these steps:

### Step 1: Simplify the Expression
First, we can simplify the expression \(1 + 3x + 3x^2 + x^3\). Notice that this can be factored as:
\[ 1 + 3x + 3x^2 + x^3 = (1 + x^3) + 3(x + x^2) = (1 + x^3) + 3x(1 + x) \]
However, a more useful simplification is to recognize that:
\[ 1 + 3x + 3x^2 + x^3 = (1 + x)^3 \]
This is because the coefficients \(1, 3, 3, 1\) correspond to the binomial coefficients in the expansion of \((1 + x)^3\).

### Step 2: Rewrite the Expression
Thus, we can rewrite the original expression:
\[ (1 + 3x + 3x^2 + x^3)^{10} = ((1 + x)^3)^{10} = (1 + x)^{30} \]

### Step 3: Apply the Binomial Theorem
Now, we can apply the Binomial Theorem to expand \((1 + x)^{30}\). The Binomial Theorem states that:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]
In our case, \(x = 1\), \(y = x\), and \(n = 30\). Therefore, the expansion will have terms of the form:
\[ \binom{30}{k} \cdot 1^{30-k} \cdot x^k = \binom{30}{k} x^k \]
for \(k = 0, 1, 2, \ldots, 30\).

### Step 4: Count the Number of Terms
The number of distinct terms in the expansion of \((1 + x)^{30}\) corresponds to the different powers of \(x\) that can occur, which are \(x^0, x^1, x^2, \ldots, x^{30}\).

Thus, the number of terms is \(30 + 1 = 31\).

### Final Answer
Therefore, the number of terms in the expansion of \((1 + 3x + 3x^2 + x^3)^{10}\) is:
\[ \boxed{31} \]