For n ≥ 2 , let S n denote the set of all subsets of { 1 , 2 … ... , n } with no two consecutive numbers. For example { 1 , 3 , 5 } ∈ S 6 , but { 1 , 2 , 4 } ∈ / S 6 . Then n ( S 5 ) is equal

Question

For n ≥ 2 , let S n ​ denote the set of all subsets of { 1 , 2 … ... , n } with no two consecutive numbers. For example { 1 , 3 , 5 } ∈ S 6 ​ , but { 1 , 2 , 4 } ∈ / S 6 ​ . Then n ( S 5 ​ ) is equal to ________

TestHub · Accepted Answer

A = { 1 , 2 , 3 , 4 , 5 … ... n } No. of subsets having r elements such that no two are consecutive is = n − r + 1 C r ​ for n = 5 , no. of ways = 6 ⋅ r C r ​ Subsets having no element = 1 Subsets having exactly 1 element = 5 C 1 ​ = 5 Subsets having exactly 2 element = 4 C 2 ​ = 6 Subsets having exactly 3 element = 3 C 3 ​ = 1 ⇒ 5 + 6 + 1 + 1 = 13

Mathematics - Permutation & Combination Question with Solution | TestHub

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