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

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We will find number of sets in which product is odd.

Number of odd elements in S = 50

Number of such subsets = 2^{50} -1 (one reduced because only non empty sets required)

Total non - empty sets possible = 2^{100} -1

Therefore, subsets A such that the product of elements in A is even = 2^{100} -1 - (2^{50}-1)

= 2^{100} -2^{50}

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