Question

Let \(S\) be the set of the first 11 natural numbers. Then the number of elements in \[ A = \{ B \subseteq S : n(B) \ge 2 \text{ and the product of all elements of } B \text{ is even} \} \] is ____________.

Hint:

To count subsets with an even product, subtract subsets made entirely of odd numbers from total subsets.

Answer 1

Correct Answer: 1979

Step 1: Define the set \(S\).
\[ S = \{1,2,3,4,5,6,7,8,9,10,11\} \] Even numbers in \(S\): 
\[ \{2,4,6,8,10\} \Rightarrow 5 \text{ elements} \] Odd numbers in \(S\): 
\[ 6 \text{ elements} \]
Step 2: Condition for product to be even. 
A product is even if the subset contains at least one even number. 

Step 3: Count all subsets with at least one even element. 
Total subsets of \(S\): 
\[ 2^{11} = 2048 \] Subsets containing only odd numbers: 
\[ 2^{6} = 64 \] So, subsets with at least one even element: 
\[ 2048 - 64 = 1984 \]
Step 4: Remove subsets with fewer than 2 elements. 
Single-element even subsets: 
\[ 5 \] Hence, required number of subsets: 
\[ 1984 - 5 = 1979 \]
Final Answer: 
\[ \boxed{1979} \]