The value of $\left|\begin{array}{lll}1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b\end{array}\right|$ is

0

It $\Delta=\left|\begin{array}{lll}1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b\end{array}\right|$

⇒  $c_3 \rightarrow c_3+c_2 \quad \Delta=\left|\begin{array}{lll}1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c\end{array}\right|$

taking (a + b + c) common factor out

$\Rightarrow(a+b+c)\left|\begin{array}{lll}1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1\end{array}\right|=(a+b+c) \times 0=0$

(two columns have same values here value is zero)