The problem asks for the number of subsets B of the set $S = \{1, 2, ..., 11\}$ such that the size of the subset, denoted as $n(B)$, is greater than or equal to 2 ($n(B) \geq 2$), and the product of all elements in B is an even number.

Set S and Subset Properties

• The given set is $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. The total number of elements in S is $n(S) = 11$.
• The total number of possible subsets of S is $2^{n(S)} = 2^{11} = 2048$.
• A product of numbers is even if at least one of the numbers is even.
• Conversely, a product of numbers is odd only if all the numbers are odd.

Calculating Subsets with Odd Product

• First, let's find the subsets where the product of elements is odd. This happens only when all elements in the subset are odd numbers.
• Identify the odd numbers in S: $O = \{1, 3, 5, 7, 9, 11\}$. The count of odd numbers is $n(O) = 6$.
• The subsets containing only odd numbers are the subsets of O. The number of such subsets is $2^{n(O)} = 2^6 = 64$.

Calculating Subsets with Even Product

• The number of subsets where the product of elements is even is the total number of subsets minus the number of subsets with an odd product.
• Number of subsets with even product = Total subsets - Number of subsets with odd product
• Number of subsets with even product = $2^{11} - 2^6 = 2048 - 64 = 1984$.

Applying the Size Condition $n(B) \geq 2$

• We need subsets with an even product AND $n(B) \geq 2$.
• The 1984 subsets with an even product include subsets of all sizes (0, 1, 2, ...).
• We need to exclude subsets from these 1984 that have $n(B) < 2$. These are subsets with size 0 or 1.
• Case 1: $n(B) = 0$ (Empty Set): The empty set $\emptyset$ has a product conventionally defined as 1 (odd). It's not included in the 1984 count.
• Case 2: $n(B) = 1$ (Singleton Sets):
  • Subsets with one odd element: $\{1\}, \{3\}, \{5\}, \{7\}, \{9\}, \{11\}$. Their product is odd. These 6 subsets are part of the 64 subsets with an odd product and are *not* included in the 1984 count.
  • Subsets with one even element: $\{2\}, \{4\}, \{6\}, \{8\}, \{10\}$. Their product is even. There are 5 such subsets. These 5 subsets *are* included in the 1984 count.
• Since the condition is $n(B) \geq 2$, we must remove the 5 singleton subsets with even elements from the count of 1984.
• The number of elements in set A = (Number of subsets with even product) - (Number of singleton subsets with even product)
• Number of elements in A = $1984 - 5 = 1979$.

Final Result

The number of subsets B of S such that $n(B) \geq 2$ and the product of elements in B is even is 1979.