Sets, Relations & Functions Question - None

Question

Given set S = \{1, 2, 3, \ldots, 50\}. Find the number of non-empty subsets of S such that the product of elements is even.

Accepted Answer

Solution:

Total number of non-empty subsets of S is 2^{50} - 1.

The product of elements in a subset is odd if and only if all elements are odd. The set of odd numbers in S is \{1, 3, 5, \ldots, 49\} which has 25 elements.

Number of non-empty subsets containing only odd numbers is 2^{25} - 1.

Therefore, the number of non-empty subsets with even product is:

(2^{50} - 1) - (2^{25} - 1) = 2^{50} - 2^{25}

∴ The answer is 2^{50} - 2^{25}.

Question:

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