The sum of all the coefficients of the terms in the expansion of `(x+y+z+w)^(6)` which contain `x` but not `y`, is

Options

A. `3^(6)`
B. `2^(6)`
C. `3^(6)-2^(6)`
D. none of these

Explanation

To solve the problem of finding the sum of all the coefficients of the terms in the expansion of \((x+y+z+w)^{6}\) which contain \(x\) but not \(y\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the sum of coefficients of terms that contain \(x\) but do not contain \(y\) in the expansion of \((x+y+z+w)^{6}\).

2. **Finding the Total Coefficients without \(y\)**:
To find the sum of coefficients of terms that do not include \(y\), we set \(y=0\) and evaluate the expression with \(x\), \(z\), and \(w\) set to \(1\):
\[ (x + 0 + z + w)^{6} = (x + z + w)^{6} \]
Now, substituting \(x=1\), \(z=1\), and \(w=1\):
\[ (1 + 1 + 1)^{6} = 3^{6} \]

3. **Finding the Total Coefficients without \(x\) and \(y\)**:
Next, we need to subtract the coefficients of terms that do not contain \(x\) and \(y\). We set \(x=0\) and \(y=0\) and evaluate the expression:
\[ (0 + 0 + z + w)^{6} = (z + w)^{6} \]
Substituting \(z=1\) and \(w=1\):
\[ (1 + 1)^{6} = 2^{6} \]

4. **Final Calculation**:
The sum of the coefficients of terms that contain \(x\) but not \(y\) is given by:
\[ \text{Sum} = 3^{6} - 2^{6} \]

5. **Calculating the Values**:
Now we can calculate \(3^{6}\) and \(2^{6}\):
\[ 3^{6} = 729 \quad \text{and} \quad 2^{6} = 64 \]
Therefore:
\[ \text{Sum} = 729 - 64 = 665 \]

### Final Answer:
The sum of all the coefficients of the terms in the expansion of \((x+y+z+w)^{6}\) which contain \(x\) but not \(y\) is \(665\).

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