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 use the formula for the number of terms in the expansion of a multinomial expression.

### Step-by-step Solution:

1. **Identify the expression**: We have the expression \( (4x^2 + 9y^2 + 12xy)^6 \). Here, we have three terms: \( 4x^2 \), \( 9y^2 \), and \( 12xy \).

2. **Count the number of distinct variables**: The variables in the expression are \( x \) and \( y \). However, we need to consider the powers of these variables in the expansion.

3. **Rewrite the expression**: We can rewrite the expression as \( (a + b + c)^6 \) where:
- \( a = 4x^2 \)
- \( b = 9y^2 \)
- \( c = 12xy \)

4. **Determine the number of distinct terms**: To find the number of distinct terms in the expansion, we need to find the number of non-negative integer solutions to the equation:
\[ 2p + 2q + r = 6 \]
where \( p \), \( q \), and \( r \) are the powers of \( a \), \( b \), and \( c \) respectively.

5. **Use the formula for the number of terms**: The number of terms in the expansion of \( (x_1 + x_2 + ... + x_p)^n \) is given by the formula:
\[ \text{Number of terms} = \binom{n + p - 1}{p - 1} \]
where \( n \) is the exponent (which is 6 in this case) and \( p \) is the number of distinct terms (which is 3 here: \( a, b, c \)).

6. **Calculate the values**: Here, \( n = 6 \) and \( p = 3 \).
\[ \text{Number of terms} = \binom{6 + 3 - 1}{3 - 1} = \binom{8}{2} \]

7. **Compute \( \binom{8}{2} \)**:
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

8. **Conclusion**: Therefore, the number of terms in the expansion of \( (4x^2 + 9y^2 + 12xy)^6 \) is **28**.