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Let A = 1, 2, 3, , n where n 5 . Define B = T A : either 1 T or 2, 3 T and C = T A : the sum of all elements of T is even . If the number of elements in the set B C is 3328 , then the value of n is _______.

Correct answer

12

Step-by-step solution

By De Morgan's law, (B C)' = B' C' . The set B' consists of subsets T of A such that 1 T and 2, 3 T . This means T must contain 1 , but cannot contain both 2 and 3 . Thus, T 1, 2, 3 can be exactly 1 , 1, 2 , or 1, 3 . The set C' consists of subsets T of A such that the sum of all elements of T is odd. Let T = T₁ T₂ , where T₁ = T 1, 2, 3 and T₂ = T 4, 5, , n . Case 1: T₁ = 1 . The sum of elements in T₁ is 1 (odd). For the total sum to be odd, the sum of elements in T₂ must be even. Case 2: T₁ = 1, 2 . The sum is 3

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