Min-coin change problem with limited coins
Let $N(t,S)$ be the least number of values from $S$ which sum to $t$. The basic recurrence is
$$
N(t, S \cup \{s\}) = \min(N(t,S),N(t-s,S)+1). 
$$
I'll let you work out the details.



This recurrence leads to an efficient algorithm only if we are allowed to run in time polynomial in $t$.

If we think of $t$ encoded in binary, then this algorithm runs in exponential time.

Indeed, the very related SUBSET-SUM problem is NP-complete.