GATE DA 2026 Set 1 — Question 19
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Probability & Statistics → Counting → Combinations
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Question
Let be a randomly chosen non-empty subset of .Which of the following is the probability that the product of all the elements of is even?
Correct answer
(C) 2¹⁰¹³(2¹⁰¹³ - 1)(2²⁰²⁶ - 1)
Solution
Let . The total number of elements in is .
The total number of non-empty subsets of is .A subset has an even product if and only if it contains at least one even element. Conversely, a subset has an odd product if and only if all its elements are odd.The odd elements in are . The number of odd elements is .
The number of non-empty subsets consisting only of odd elements is .Therefore, the number of non-empty subsets with at least one even element is:
.The probability is the ratio of this count to the total number of non-empty subsets:
.Thus, option (C) is correct.
The total number of non-empty subsets of is .A subset has an even product if and only if it contains at least one even element. Conversely, a subset has an odd product if and only if all its elements are odd.The odd elements in are . The number of odd elements is .
The number of non-empty subsets consisting only of odd elements is .Therefore, the number of non-empty subsets with at least one even element is:
.The probability is the ratio of this count to the total number of non-empty subsets:
.Thus, option (C) is correct.
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