Let S be the set of the first 11 natural numbers. Then the number of elements in A= B S: n(B) 2 and the product of all elements of B is even is _ _ _ _ .

Step-by-step solution

The set S = 1, 2, 3, ..., 11 has 11 elements. We need to count subsets B with n(B) 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) 2 are all except empty set (1) and singletons (6): 64 - 7 = 57 . Total subsets with n(B) 2 : 2¹¹ - 1 - 11 = 2036 . Subsets with even product and n(B) 2 : 2036 - 57 = 19