Find `|{:(1+a,b,c),(a,1+b,c),(a,b,1+c):}|`

Answer

To find the determinant \( | \begin{pmatrix} 1 + a & b & c \\ a & 1 + b & c \\ a & b & 1 + c \end{pmatrix} | \), we will follow these steps:

### Step 1: Write the determinant
We start with the determinant:
\[ D = \begin{vmatrix} 1 + a & b & c \\ a & 1 + b & c \\ a & b & 1 + c \end{vmatrix} \]

### Step 2: Simplify the first column
We can simplify the first column by adding the second and third columns to it. This gives us:
\[ D = \begin{vmatrix} 1 + a + b + c & b & c \\ a + (1 + b) + c & 1 + b & c \\ a + b + (1 + c) & b & 1 + c \end{vmatrix} \]
This can be rewritten as:
\[ D = \begin{vmatrix} 1 + a + b + c & b & c \\ 1 + a + b + c & 1 + b & c \\ 1 + a + b + c & b & 1 + c \end{vmatrix} \]

### Step 3: Factor out the common term
Notice that the first column has a common factor of \( 1 + a + b + c \). We can factor this out:
\[ D = (1 + a + b + c) \begin{vmatrix} 1 & b & c \\ 1 & 1 + b & c \\ 1 & b & 1 + c \end{vmatrix} \]

### Step 4: Perform row operations
Next, we can perform row operations to simplify the determinant:
- Subtract the first row from the second and third rows:
\[ D = (1 + a + b + c) \begin{vmatrix} 1 & b & c \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} \]

### Step 5: Calculate the determinant
Now, the determinant of the resulting matrix is:
\[ D = (1 + a + b + c) \cdot (1 \cdot 1 \cdot 1) = 1 + a + b + c \]

### Final Answer
Thus, the value of the determinant is:
\[ \boxed{1 + a + b + c} \]