In the question, it is given to us that: $c_1{}, ..., c_n{}$ are scalers and not all are zero.
$\sum_{i=1}^{n} c_{i}a_{i} = 0$ (Given)
$\Rightarrow$ $c_1a_1{} + c_2a_2{} + … + c_na_n = 0$
This means the Linear Combination of above $a_i$ results in 0 and there is at least one $c_i$ which is non-zero.
We can infer that the vectors $a_i$ are thus Linearly dependent. $...(1)$
Considering $Ax = b$, where $A = [a_1, a_2, …, a_n]$ and $b = \sum_{i=1}^{n} a_{i}$
Upon expanding the $b = \sum_{i=1}^{n} a_{i}$
$\Rightarrow$ $b = a_1{} + a_2{} + … + a_n$ $...(2)$
But since, from (1) we inferred that vectors $a_i$ are Linearly dependent, we can say that for (2) it can be infinitely many possible ways.
So the correct answer is (C).