Let $S=\{1,2,3,\dots ,100\}$. The number of non-empty subsets$A$ of $S$ such that the product of elements in $A$ is even is 

Here, $S=\{1,2,3,\dots .100\}$  

So the number of non- empty subsets $A$of $S$ such that the product of elements in $A$ is even 

= Total non empty subsets - subsets with product of element is odd 

=$2^{100} -1-\left|2^{50}-1\right|=2^{100}-2^{50}=2^{50}\left(2^{50}-1\right)$