Find the number of terms in the expansion of `[ ( 2x - 3y )^(2) ( 2x + 3y )^(2) ]^(2) `?

Options

A. 4
B. 5
C. 8
D. 16

Answer

To find the number of terms in the expansion of \(\left( (2x - 3y)^2 (2x + 3y)^2 \right)^2\), we can follow these steps:

### Step 1: Simplify the Expression
First, we can rewrite the expression:
\[ \left( (2x - 3y)^2 (2x + 3y)^2 \right)^2 = \left( (2x - 3y)(2x + 3y) \right)^4 \]
This is because \((a^2)(b^2) = (ab)^2\).

### Step 2: Use the Difference of Squares
Next, we can apply the difference of squares formula:
\[ (2x - 3y)(2x + 3y) = (2x)^2 - (3y)^2 = 4x^2 - 9y^2 \]
Thus, we have:
\[ \left( 4x^2 - 9y^2 \right)^4 \]

### Step 3: Determine the Number of Terms
Now, we need to find the number of distinct terms in the expansion of \((4x^2 - 9y^2)^4\). According to the Binomial Theorem, the number of terms in the expansion of \((a + b)^n\) is given by \(n + 1\).

Here, \(n = 4\), so the number of terms is:
\[ 4 + 1 = 5 \]

### Conclusion
Therefore, the number of terms in the expansion of \(\left( (2x - 3y)^2 (2x + 3y)^2 \right)^2\) is **5**.

---