Let `S={1,2,3, …, 100}`. The number of non-empty subsets A to S such that the product of elements in A is even is

Options

A. `2^(50)(2^(50) - 1)`
B. `2^(100)-1`
C. `2^(50)-1`
D. `2^(50) + 1`

Answer

To solve the problem, we need to find the number of non-empty subsets \( A \) of the set \( S = \{1, 2, 3, \ldots, 100\} \) such that the product of the elements in \( A \) is even.

### Step-by-step Solution:

1. **Determine the Total Number of Subsets**:
The total number of subsets of a set with \( n \) elements is given by \( 2^n \). Since our set \( S \) has 100 elements, the total number of subsets is:
\[ 2^{100} \]

2. **Calculate the Number of Non-empty Subsets**:
The total number of non-empty subsets is given by subtracting the empty set from the total number of subsets:
\[ 2^{100} - 1 \]

3. **Identify Odd and Even Numbers in the Set**:
In the set \( S \), there are 50 even numbers (2, 4, 6, ..., 100) and 50 odd numbers (1, 3, 5, ..., 99).

4. **Calculate the Number of Subsets with Odd Products**:
A subset will have an odd product if it contains only odd numbers. The number of subsets that can be formed using the 50 odd numbers is:
\[ 2^{50} \]
This includes the empty subset, so the number of non-empty subsets with an odd product is:
\[ 2^{50} - 1 \]

5. **Calculate the Number of Subsets with Even Products**:
To find the number of non-empty subsets where the product is even, we subtract the number of non-empty subsets with an odd product from the total number of non-empty subsets:
\[ \text{Number of non-empty subsets with even product} = (2^{100} - 1) - (2^{50} - 1) \]
Simplifying this gives:
\[ = 2^{100} - 1 - 2^{50} + 1 = 2^{100} - 2^{50} \]

### Final Answer:
Thus, the number of non-empty subsets \( A \) of \( S \) such that the product of elements in \( A \) is even is:
\[ 2^{100} - 2^{50} \]