The subset sum problem and subset product problem are NP-complete. Is the following problem polynomial-time solvable: given a set of positive integers, find a subset whose sum is $S$ and whose product is $T$.
It is still NP-complete. Here is a very sketchy reduction from subset sum. The goal of the whole reduction will be to make $T$ unimportant. If the inputs for subset sum are $a_i$, then add $a_i$ and $2^Na_i$ to our inputs, where $N$ is large, but still $poly(n)$. Set $S$ to be $2^N$ times the original subset sum plus $2^N-1$. Set $T$ to be the product of the $a_i$'s multiplied by a sufficiently big power of $2$, to be determined later. The idea is that if the number with index $i$ was not picked, then we can pick $a_i$, while if it was picked, we can pick $2^Na_i$. Finally, add a lot of each small power of $2$ to our input to be able to make the desired sum and product. Notice that this gives us enough liberty to get $S$ and $T$.