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Solution


Q.

For n≥2, let Sn denote the set of all subsets of {1, 2, ..., n} with no two consecutive numbers. For example {1,3,5}∈S6 but {1,2,4}∉S6. Then n(S5) is equal to __________.          [2025]

Ans.

(13)

Let A = {1, 2, 3, 4, 5, ..., n}

Number of subsets having r elements such that no two are consecutive = Crn–r+1

For n = 5, number of ways = Cr6–r

Subsets having no elements = 1 i.e., ϕ

Subsets having exactly 1 element = C15 = 5 i.e., {1}, {2}, {3}, {4}, {5}

Subsets having exactly 2 elements = C24 = 6 i.e., {1, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 5}

Subsets having exactly 3 elements = C33 = 1 i.e., {1, 3, 5}

∴  n(S5) = 1 + 5 + 6 + 1 = 13