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 ________

Step-by-step solution

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