Start by observing what is important — the number of coins of each denomination. Now it's easy to note that if you don't have a coin of denomination $$$1$$$ then $$$1$$$ is your answer right away.

For other cases you can do the following — Build a map, $$$fre$$$ that stores frequency of each denomination. Iterate over all the values, keep a track of prefix sums i.e. $$$pref[i]$$$ denotes $$$\sum j*fre[j]$$$ for $$$j \le i$$$. Suppose the current coin value is $$$i$$$, then the claim is if you can create all the sums $$$<i$$$, then you can create all the sums up to $$$pref[i]$$$. Now just check that the next denomination you encounter does not exceed this value by more than $$$1$$$.

The claim can be simply proven by induction.