Sum of coefficients in the expansion of `(x+2y+z)^(10)` is

Options

A. `2^(10)`
B. `3^(10)`
C. 1
D. none of these

Answer

To find the sum of the coefficients in the expansion of \((x + 2y + z)^{10}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We want to find the sum of the coefficients in the polynomial expansion of \((x + 2y + z)^{10}\). The coefficients are the numerical factors in front of each term when the polynomial is fully expanded.

2. **Using the Trick**: A useful trick to find the sum of the coefficients in a polynomial is to substitute all variables with 1. This means we will set \(x = 1\), \(y = 1\), and \(z = 1\).

3. **Substituting Values**: Substitute \(x\), \(y\), and \(z\) in the expression:
\[ (1 + 2 \cdot 1 + 1)^{10} \]

4. **Calculating Inside the Parenthesis**: Simplify the expression inside the parenthesis:
\[ 1 + 2 \cdot 1 + 1 = 1 + 2 + 1 = 4 \]

5. **Raising to the Power**: Now raise the result to the power of 10:
\[ 4^{10} \]

6. **Final Calculation**: Calculate \(4^{10}\):
\[ 4^{10} = (2^2)^{10} = 2^{20} = 1048576 \]

### Conclusion:
The sum of the coefficients in the expansion of \((x + 2y + z)^{10}\) is \(1048576\).

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