The sum of the numerical coefficients in the expansion of `(1+x/3+(2y)/3)^12`, is

Options

A. 1
B. 2
C. `2^(12)`
D. none of these

Explanation

To find the sum of the numerical coefficients in the expansion of \( (1 + \frac{x}{3} + \frac{2y}{3})^{12} \), we can follow these steps:

### Step 1: Understand the expression
The expression we are dealing with is \( (1 + \frac{x}{3} + \frac{2y}{3})^{12} \). The coefficients in the expansion can be found using the Binomial Theorem.

### Step 2: Use the Binomial Theorem
According to the Binomial Theorem, the expansion of \( (a + b + c)^n \) can be expressed in terms of combinations. The coefficients of the expansion are given by the multinomial coefficients.

### Step 3: Find the sum of coefficients
To find the sum of the numerical coefficients in the expansion, we can substitute \( x = 1 \) and \( y = 1 \) into the expression. This is because the sum of the coefficients is equivalent to evaluating the expression at these values.

### Step 4: Substitute values
Substituting \( x = 1 \) and \( y = 1 \) into the expression gives:
\[ 1 + \frac{1}{3} + \frac{2 \cdot 1}{3} = 1 + \frac{1}{3} + \frac{2}{3} = 1 + 1 = 2 \]

### Step 5: Raise to the power of 12
Now we raise the result to the power of 12:
\[ (2)^{12} \]

### Step 6: Final result
Thus, the sum of the numerical coefficients in the expansion is:
\[ 2^{12} \]

### Conclusion
The answer is \( 2^{12} \).

---