Let X be a set containing n elements. The no. of all the ordered triplets ( A , B , C ) such that C is a subset of B and B is a proper subset of A where A ⊆ X , is

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C ⊆ B ⊂ A ⊆ X The no. of ways = ∑ i = 0 n ​ n C i ​ [ ∑ j = 0 i − 1 ​ i C j ​ [ ∑ k = 0 j ​ j C k ​ ] ] = ∑ i = 0 n ​ n C i ​ ∑ j = 0 i − 1 ​ i C j ​ 2 j = ∑ i = 0 n ​ n C i ​ ( 3 i − 2 i ) = 4 n − 3 n

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JEE Main 2027 (Class 12)