The number of terms in the expansion of ` (4x^(2) + 9y^(2) + 12xy)^(6) ` is

Options

A. 2
B. 12
C. 13
D. 26

Answer

To find the number of terms in the expansion of \( (4x^2 + 9y^2 + 12xy)^6 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the expression**: We start with the expression \( (4x^2 + 9y^2 + 12xy)^6 \).

2. **Recognize the form**: Notice that \( 4x^2 + 9y^2 + 12xy \) can be rewritten. We can factor it as:
\[ (2x + 3y)^2 \]
Thus, we can rewrite the original expression as:
\[ ((2x + 3y)^2)^6 = (2x + 3y)^{12} \]

3. **Use the Binomial Theorem**: According to the Binomial Theorem, the expansion of \( (a + b)^n \) has \( n + 1 \) terms. Here, \( n = 12 \).

4. **Calculate the number of terms**: Since we have \( (2x + 3y)^{12} \), the number of terms in this expansion is:
\[ 12 + 1 = 13 \]

Thus, the number of terms in the expansion of \( (4x^2 + 9y^2 + 12xy)^6 \) is **13**.