Let S be the set of the first 11 natural numbers. Then the number of elements in $$ A= \ B \subseteq S:n(B)\geq 2 $$, and the product of all elements of B is even} is __________.

$$S = \{1,...,11\}$$. Subsets B with $$|B| \geq 2$$ and product of elements is even (at least one even element).

Total subsets with $$|B| \geq 2$$: $$2^{11}-1-11 = 2048-12 = 2036$$.

Subsets with $$|B| \geq 2$$ and all odd: odd elements in S: $$\{1,3,5,7,9,11\}$$ (6 elements). Subsets of size $$\geq 2$$: $$2^6-1-6 = 57$$.

Answer = $$2036-57 = 1979$$.

The answer is 1979.