Let M be a randomly chosen non-empty subset of {1,2,3,..,2026}. Which of the following is a probability that product all the elements of M is even.

The correct option is:
$\frac{2^{1013}(2^{1013}-1)}{(2^{2026}-1)}$

Step-by-Step Explanation:

1. Understand the Sample Space:
We are given a set containing the first 2026 positive integers:
S = {1, 2, 3, ..., 2026}
The number of elements in this set is 2026. A non-empty subset M is chosen at random.
The total number of possible subsets of a set of size 2026 is 2²⁰²⁶. Since M must be non-empty, we exclude the empty set. Thus, the total number of non-empty subsets (which is our sample space) is:
Total non-empty subsets = 2²⁰²⁶ - 1

2. Determine the Condition for an Even Product:
The product of the elements in a subset M is even if and only if there is at least one even number in the subset.
Conversely, the product is odd if and only if all the elements in the subset are odd numbers.
To find the number of subsets with an even product, we can subtract the number of subsets with an all-odd product from the total number of non-empty subsets.

3. Count the Odd Numbers and Subsets:
In the set {1, 2, 3, ..., 2026}, exactly half of the numbers are even and half are odd.
Number of odd numbers = 2026 / 2 = 1013
Number of even numbers = 1013
The subsets of S that contain only odd numbers must be chosen entirely from this set of 1013 odd numbers.
The total number of non-empty subsets consisting of only odd numbers is:
Subsets with all-odd elements = 2¹⁰¹³ - 1

4. Calculate the Favorable Outcomes:
The number of non-empty subsets M with an even product is:
Favorable subsets = (Total non-empty subsets) - (Non-empty subsets with all-odd elements)
Favorable subsets = (2²⁰²⁶ - 1) - (2¹⁰¹³ - 1)
Favorable subsets = 2²⁰²⁶ - 2¹⁰¹³
By factoring out 2¹⁰¹³, we get:
Favorable subsets = 2¹⁰¹³(2¹⁰¹³ - 1)

5. Compute the Probability:
The probability that the product of all elements of M is even is the ratio of favorable subsets to the total number of non-empty subsets:
Probability = Favorable subsets / Total non-empty subsets
$\text{Probability}=\frac{2^{1013}(2^{1013}-1)}{(2^{2026}-1)}$