c
Condition for non-trivial solution is
$\left|\begin{array}{ccc}{a-1} & {a} & {a} \\ {b} & {b-1} & {b} \\ {c} & {c} & {c-1}\end{array}\right|=0$
$\Rightarrow(a+b+c-1)\left|\begin{array}{ccc}{1} & {1} & {1} \\ {b} & {b-1} & {b} \\ {c} & {c} & {c-1}\end{array}\right|=0$
$\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{1}$
$\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\mathrm{C}_{1}$
$\Rightarrow a+b+c=1$