Power Sum

Chef is preparing a brand new dish. He found $N$ new ingredients (numbered $1$ through $N$) for the dish. For each valid $i$, the $i$-th ingredient costs $C_i$ dollars and it has value $V_i$. Chef has a budget: $S$ dollars. He can buy each ingredient at most once. A non-empty subset of these $N$ ingredients (possibly containing all ingredients) is called *affordable* if the sum of costs of all th