No. of term in `(1 + 3x + 3x^2 + x^3)^6` is

Options

A. 17
B. 19
C. 21
D. 16

Answer

To find the number of terms in the expression \((1 + 3x + 3x^2 + x^3)^6\), we can follow these steps:

### Step 1: Identify the polynomial
The expression inside the parentheses is a polynomial:
\[ P(x) = 1 + 3x + 3x^2 + x^3 \]

### Step 2: Determine the degree of the polynomial
The highest degree of \(P(x)\) is 3 (from \(x^3\)).

### Step 3: Use the binomial theorem
The expression can be rewritten using the binomial theorem. We can express \(P(x)\) as:
\[ P(x) = (x + 1)^3 \]
This is because the coefficients \(1, 3, 3, 1\) correspond to the binomial coefficients of \((x + 1)^3\).

### Step 4: Raise the polynomial to the power of 6
Now, we need to raise this polynomial to the power of 6:
\[ (P(x))^6 = ((x + 1)^3)^6 = (x + 1)^{18} \]

### Step 5: Find the number of terms in the expanded polynomial
The number of terms in the expansion of \((x + 1)^{18}\) is given by the formula:
\[ \text{Number of terms} = n + 1 \]
where \(n\) is the exponent. In this case, \(n = 18\).

### Step 6: Calculate the number of terms
Thus, the number of terms is:
\[ 18 + 1 = 19 \]

### Conclusion
The number of terms in the expression \((1 + 3x + 3x^2 + x^3)^6\) is **19**.

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