Let S be the set of the first 11 natural numbers. Then the number...

The set $S = \{1, 2, 3, ..., 11\}$ has 11 elements.
We need to count subsets $B$ with $n(B) \geq 2$ and even product.
A product is even iff at least one element is even.
Equivalently, count complements: a product is odd iff all elements are odd.
The odd numbers in $S$ are $\{1, 3, 5, 7, 9, 11\}$ (6 elements).
Subsets of odd numbers: $2^6 = 64$.
Among these, those with $n(B) \geq 2$ are all except empty set (1) and singletons (6): $64 - 7 = 57$.
Total subsets with $n(B) \geq 2$: $2^{11} - 1 - 11 = 2036$.
Subsets with even product and $n(B) \geq 2$: $2036 - 57 = 1979$.