Option A is correct
A product of integers is even if and only if at least one of the integers in the set is even.
Conversely, a product is odd only if every single number in the set is odd.
Breaking Down the Set $S$
The set $S = \{1, 2, 3, \dots, 100\}$ contains:
50 Even numbers: $\{2, 4, 6, \dots, 100\}$
50 Odd numbers: $\{1, 3, 5, \dots, 99\}$
Method 1: Subtraction (Total - Undesirable)
Total non-empty subsets of $S$:
Since there are 100 elements, the total number of subsets is $2^{100}$. Excluding the empty set,
$$\text{Total} = 2^{100} - 1$$
Subsets with an ODD product:
To get an odd product, we must pick elements only from the 50 odd numbers.
The number of non-empty subsets using only odd numbers is:
$$\text{Odd Products} = 2^{50} - 1$$
Subsets with an EVEN product:
$$\text{Even Products} = (2^{100} - 1) - (2^{50} - 1)$$
$$\text{Even Products} = 2^{100} - 1 - 2^{50} + 1 = 2^{100} - 2^{50}$$
Method 2: Direct Construction
Pick the even numbers: To ensure the product is even, we must pick at least one even number from the 50 available.
Number of ways = $2^{50} - 1$ (Total even subsets minus the empty even set).
Pick the odd numbers: We can pick any combination of the 50 odd numbers (including none at all), because the presence of an even number already guarantees an even product.
Number of ways = $2^{50}$.
Even product: $(2^{50} - 1) \times 2^{50}$
Why the answer is NOT $2^{50} \times 2^{50}$ is not Correct
The value $2^{50} \times 2^{50}$ is equal to $2^{100}$.It includes the Empty Set: $2^{100}$ includes the case where you pick zero even numbers and zero odd numbers. The question specifically asks for non-empty subsets.
