1. We need the number of non-empty subsets $A \subseteq S$ such that the product of elements of $A$ is even, where $S=\{1,2,3,\dots,100\}.$

2. A product is even if and only if at least one even number is present in the subset.

So, we count: $\text{Required subsets} = \text{all non-empty subsets} - \text{non-empty subsets containing only odd numbers}.$

3. Total number of elements in $S$ is $100$. Hence total number of subsets is $2^{100}.$ Therefore total number of non-empty subsets is $2^{100}-1.$

4. Now count the odd numbers in $S$. From $1$ to $100$, exactly half are odd, so number of odd elements is $50.$

Thus, the number of subsets containing only odd numbers is $2^{50}.$ Among these, the number of non-empty subsets containing only odd numbers is $2^{50}-1.$

5. Therefore, $\text{Required number}=(2^{100}-1)-(2^{50}-1).$ Simplifying, $\text{Required number}=2^{100}-2^{50}.$

6. Factorizing, $2^{100}-2^{50}=2^{50}(2^{50}-1).$

7. Now compare with the options:

• A: $2^{50}-1$
• B: $2^{50}(2^{50}-1)$
• C: $2^{100}-1$
• D: $2^{50}+1$

Hence the correct option is $\boxed{B}.$