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 (a) `3^(6)` (b) `2^(6)` (c) `3^(6)-2^(6)` (d) none of these

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 \((x+y+z+w)^{6}\) that contain \(x\) but not \(y\), we can follow these steps:

### Step 1: Find the sum of coefficients when \(y\) is not included.
To find the sum of coefficients of the expansion that does not include \(y\), we can substitute \(y = 0\) and set \(x = z = w = 1\). This gives us:
\[ (1 + 0 + 1 + 1)^{6} = (3)^{6} \]

### Step 2: Find the sum of coefficients when both \(x\) and \(y\) are not included.
Next, we need to find the sum of coefficients where both \(x\) and \(y\) are not included. We substitute \(x = 0\), \(y = 0\), and set \(z = w = 1\). This gives us:
\[ (0 + 0 + 1 + 1)^{6} = (2)^{6} \]

### Step 3: Calculate the required sum.
The sum of all coefficients of the terms that contain \(x\) but not \(y\) can be found by subtracting the sum of coefficients where both \(x\) and \(y\) are not included from the sum of coefficients where \(y\) is not included:
\[ \text{Sum} = (3^{6}) - (2^{6}) \]

### Conclusion
Thus, the sum of all the coefficients of the terms in the expansion of \((x+y+z+w)^{6}\) that contain \(x\) but not \(y\) is:
\[ 3^{6} - 2^{6} \]

The correct answer is option (c) \(3^{6} - 2^{6}\).

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