Let A = 1 , 2 , 3 , … . 7 and let P ( A ) denote the power set of A . If the number of functions f : A → P ( A ) such that a ∈ f ( a ) , ∀ a ∈ A is m n , m and n ∈ N and m is least, then m + n is equal to ______.

Step-by-step solution

Given, A = 1 , 2 , 3 , … . 7 And f : A → P ( A ) and a ∈ A Also a ∈ f ( a ) , that means ‘ a ’ will be the element of subset as f : A → P ( A ) , Now, total options for 1 will be 2 6 . (Because 2 6 subsets contains elements 1 ) Similarly, for every other element we get, total 2 6 × 2 6 × 2 6 × 2 6 × 2 6 × 2 6 × 2 6 = 2 42 subsets. Now, comparing with m n we get, m = 2 & n = 42 Hence, m + n = 2 + 42 = 44