The number of terms in the expansion of `[(x+2y)^(4) xx (x-2y)^(4) ]^(2)` are

Options

A. 8
B. 7
C. 9
D. 50

Answer

To find the number of terms in the expansion of \([(x + 2y)^{4} \cdot (x - 2y)^{4}]^{2}\), we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression \([(x + 2y)^{4} \cdot (x - 2y)^{4}]^{2}\). We can first simplify the product inside the brackets.

Using the identity \( (a + b)(a - b) = a^2 - b^2 \):
\[ (x + 2y)(x - 2y) = x^2 - (2y)^2 = x^2 - 4y^2 \]

Thus, we can rewrite the expression as:
\[ [(x^2 - 4y^2)^{4}]^{2} \]

### Step 2: Apply the Power of a Power Rule
Next, we apply the power of a power rule:
\[ [(x^2 - 4y^2)^{4}]^{2} = (x^2 - 4y^2)^{8} \]

### Step 3: Determine the Number of Terms
Now, we need to find the number of terms in the expansion of \((x^2 - 4y^2)^{8}\).

According to the Binomial Theorem, the number of terms in the expansion of \((a + b)^{n}\) is given by \(n + 1\). Here, \(n = 8\).

Thus, the number of terms is:
\[ 8 + 1 = 9 \]

### Final Answer
Therefore, the number of terms in the expansion of \([(x + 2y)^{4} \cdot (x - 2y)^{4}]^{2}\) is **9**.

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