Is it possible to solve the coin denomination problem using a 1-D array?
It is perfectly possible to solve the problem using a 1D array. Actually, you can even reconstruct the solution using just that 1D array. There are often multiple ways to solve a given problem, and in this case there's a neat time-memory tradeoff: in the 1D example you need $O(m)$ time to compute the value of one array slot, in the 2D example you only need $O(1)$ time but there are $O(nm)$ slots rather than just $O(n)$.