$\text{ Let }P=\{1,2,3,\dots ,10\}$ be the number of non-empty subsets A of S such that the product of elements in A is even is

Clearly required number of subsets of S is equal to the total number of non-empty subsets of $P$ minus the number of non-empty subsets of $S$ containing only odd natural numbers in $S.$

Required number of subsets $=\left(2^{100}-1\right)-\left(2^{50}-1\right)=2^{50}\left(2^{50}-1\right).$