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

2^1013(2^1013 - 1)/(2^2026 - 1)

GATE DA 2026 Q19: Probability a random subset of {1..2026} has an even product

Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \ldots, 2026\}$ .

Which of the following is the probability that the product of all the elements of $M$ is even?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: C (official GATE 2026 final answer key)

Concept used: Complement counting: a product is odd only if every element is odd.

First step: There are $2^{2026} - 1$ non-empty subsets; subsets of the 1013 odd numbers give odd products, all others have an even element.

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key