Sum of the coefficients of `(1+ x/3 + (2y)/(3))^12`

Options

A. `2^6`
B. `2^8`
C. `2^12`
D. `10^2`

Answer

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

### Step-by-Step Solution:

1. **Identify the expression**: We have the expression \((1 + \frac{x}{3} + \frac{2y}{3})^{12}\).

2. **Substitute variables with 1**: To find the sum of the coefficients, we substitute \(x = 1\) and \(y = 1\) into the expression. This gives us:
\[ (1 + \frac{1}{3} + \frac{2 \cdot 1}{3})^{12} \]

3. **Simplify the expression**: Now, we simplify the expression inside the parentheses:
\[ 1 + \frac{1}{3} + \frac{2}{3} = 1 + \frac{1 + 2}{3} = 1 + 1 = 2 \]

4. **Raise to the power of 12**: Now we raise the simplified result to the power of 12:
\[ 2^{12} \]

5. **Final answer**: Thus, the sum of the coefficients of the expression \((1 + \frac{x}{3} + \frac{2y}{3})^{12}\) is:
\[ 2^{12} \]

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

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