The number of terms in expansion of `(x^(2)+18x+81)^(15)` is

Options

A. 15
B. 16
C. 30
D. 31

Answer

To find the number of terms in the expansion of \((x^2 + 18x + 81)^{15}\), we can follow these steps:

### Step 1: Simplify the expression
First, we need to simplify the expression \(x^2 + 18x + 81\). We can factor this quadratic expression.

\[ x^2 + 18x + 81 = (x + 9)^2 \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression using the factored form:

\[ (x^2 + 18x + 81)^{15} = ((x + 9)^2)^{15} \]

### Step 3: Apply the power of a power rule
Using the power of a power property, we can simplify this further:

\[ ((x + 9)^2)^{15} = (x + 9)^{30} \]

### Step 4: Determine the number of terms in the binomial expansion
Now we need to find the number of terms in the expansion of \((x + 9)^{30}\). According to the Binomial Theorem, the number of terms in the expansion of \((a + b)^n\) is given by \(n + 1\).

Here, \(n = 30\):

\[ \text{Number of terms} = 30 + 1 = 31 \]

### Final Answer
Thus, the number of terms in the expansion of \((x^2 + 18x + 81)^{15}\) is **31**.

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