The minimum number of coins the sum of which is S

Question

Given a list of N coins, their values (V1, V2, ... , VN), and the total sum S. Find the minimum number of coins the sum of which is S (we can use as many coins of one type a

Answer 1

I think the approach you want is like this:

You know that you want to produce a sum S. The only ways to produce S are to first produce S-V1, and then add a coin of value V1; or to produce S-V2 and then add a coin of value V2; or...

In turn, T=S-V1 is producible from T-V1, or T-V2, or...

By stepping back in this way, you can determine the best way, if any, to produce S from your Vs.