Question

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:
(A) 2¹⁰⁰ – 1 (B) 2⁵⁰ (2⁵⁰ – 1)
(C) 2⁵⁰ – 1 (D) 2⁵⁰ + 1

Answer 1

Answer:

it's easy to calculate total subsets with product of elements to be odd..

we have a set {1,3,5,...,99}.

here total possible subsets are

${2}^{ 50}$

therefore for even subtract it from total no. of subsets

${2}^{100}$

so answer is (B)..

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Answer 2

2^50(2^50 -1)

Step-by-step explanation:

total no. of nonempty subsets = 2^100

No. of nonempty subsets with product of elements is odd = 2^50 - 1

No. of non empty subsets with product of elements is even = (2^100 - 1) - ( 2^50 - 1 )

= 2^100 - 2^50

= 2^50(2^50 - 1)