Key Concepts and Formulas

• Total Number of Subsets: A set with $n$ elements has $2^n$ subsets. The number of non-empty subsets is $2^n - 1$.
• Parity of a Product: The product of a set of integers is odd if and only if all the integers in the set are odd. If at least one integer is even, the product is even.
• Complementary Counting: The number of elements in a set $A$ that satisfy a property $P$ can be found by subtracting the number of elements that do not satisfy $P$ from the total number of elements. That is, $|A \text{ with } P| = |\text{Total}| - |A \text{ without } P|$.

Step-by-Step Solution

Step 1: Analyze the given set $S$ and identify odd and even numbers. The set is $S = \{1, 2, 3, \ldots, 100\}$. The total number of elements in $S$ is $n(S) = 100$. We need to separate the odd and even numbers in $S$. The set of odd numbers in $S$ is $O = \{1, 3, 5, \ldots, 99\}$. The number of odd elements is $n(O) = \frac{99 - 1}{2} + 1 = \frac{98}{2} + 1 = 49 + 1 = 50$. The set of even numbers in $S$ is $E = \{2, 4, 6, \ldots, 100\}$. The number of even elements is $n(E) = \frac{100 - 2}{2} + 1 = \frac{98}{2} + 1 = 49 + 1 = 50$. Alternatively, $n(E) = n(S) - n(O) = 100 - 50 = 50$.

Step 2: Determine the total number of non-empty subsets of $S$. The total number of subsets of a set with 100 elements is $2^{100}$. Since we are looking for non-empty subsets, we exclude the empty set. Total number of non-empty subsets of $S = 2^{100} - 1$.

Step 3: Identify the condition for the product of elements in a subset to be odd. The product of elements in a subset $A$ is odd if and only if all elements in $A$ are odd. This means that any subset whose product is odd must be formed entirely from the odd numbers present in $S$.

Step 4: Calculate the number of non-empty subsets with an odd product. Subsets with an odd product can only be formed using elements from the set of odd numbers $O = \{1, 3, \ldots, 99\}$, which has 50 elements. The total number of subsets that can be formed using these 50 odd numbers is $2^{50}$. Since we need non-empty subsets, we subtract 1 (for the empty set). The number of non-empty subsets of $S$ whose product of elements is odd is $2^{50} - 1$.

Step 5: Use complementary counting to find the number of non-empty subsets with an even product. We want to find the number of non-empty subsets where the product of elements is even. This is the complement of subsets where the product is odd. Number of non-empty subsets with even product = (Total number of non-empty subsets) - (Number of non-empty subsets with odd product) Number of non-empty subsets with even product = $(2^{100} - 1) - (2^{50} - 1)$ Number of non-empty subsets with even product = $2^{100} - 1 - 2^{50} + 1$ Number of non-empty subsets with even product = $2^{100} - 2^{50}$ We can factor this expression: Number of non-empty subsets with even product = $2^{50}(2^{50} - 1)$.

Common Mistakes & Tips

• Forgetting "non-empty": Always remember to subtract 1 from the total number of subsets if the question specifies "non-empty".
• Misapplying parity rules: Ensure you correctly understand that a product is odd only if all factors are odd. The presence of even a single even number makes the product even.
• Confusing the set of odd/even numbers: Double-check the counts of odd and even numbers within the given range.

Summary

To find the number of non-empty subsets of $S = \{1, 2, \ldots, 100\}$ where the product of elements is even, we used the principle of complementary counting. We first determined the total number of non-empty subsets. Then, we calculated the number of non-empty subsets whose product is odd, which are precisely those subsets formed entirely from the odd numbers in $S$. Subtracting the latter from the former gave us the desired count. The set $S$ contains 50 odd numbers and 50 even numbers. The total number of non-empty subsets is $2^{100} - 1$. The number of non-empty subsets with an odd product is $2^{50} - 1$. Therefore, the number of non-empty subsets with an even product is $(2^{100} - 1) - (2^{50} - 1) = 2^{100} - 2^{50} = 2^{50}(2^{50} - 1)$.

The final answer is $\boxed{2^{50} - 1}$.