Consider the expansion of `(a+b+c+d)^(6)`. Then the sum of all the coefficients of the term
Which contains a but not b is (a) 729 (b) 3367 (c) 665 (d) 1024

Options

A. 729
B. 3367
C. 665
D. 1024

Explanation

To find the sum of all the coefficients of the terms in the expansion of \((a+b+c+d)^6\) that contain \(a\) but not \(b\), we can follow these steps:

### Step 1: Identify the relevant terms
We need to consider the terms that contain \(a\) and can include \(c\) and \(d\), but must not include \(b\). Thus, we are essentially looking at the distribution of the remaining variables \(a\), \(c\), and \(d\).

### Step 2: Count the total distributions including \(a\)
When we include \(a\), we can think of the remaining \(5\) objects (since one \(a\) is already chosen) being distributed among the three variables \(a\), \(c\), and \(d\). The total number of ways to distribute \(6\) objects into \(3\) boxes (where \(a\) is mandatory) is given by:
\[ 3^6 \]

### Step 3: Count the distributions excluding \(a\)
Next, we need to subtract the cases where \(a\) is not included at all. In this case, we only have \(c\) and \(d\) to fill the \(6\) objects. The number of ways to distribute \(6\) objects into \(2\) boxes (only \(c\) and \(d\)) is given by:
\[ 2^6 \]

### Step 4: Calculate the values
Now, we can compute the values:
\[ 3^6 = 729 \]
\[ 2^6 = 64 \]

### Step 5: Find the required sum of coefficients
Now, we subtract the two results to find the sum of the coefficients of the terms that contain \(a\) but not \(b\):
\[ 3^6 - 2^6 = 729 - 64 = 665 \]

### Conclusion
Thus, the sum of all the coefficients of the terms that contain \(a\) but not \(b\) is:
\[ \boxed{665} \]