?19E Show that ?x! ? is not O? ?(2?? .

Solution: Step-1: n In this problem we need to show that (n!) is not O(2 ). Note: Let us consider f and g are functions from the set of integers to the set of real numbers. The estimate value can be said that f(x) is O(g(x)) if there are constants C and k such that |f(x)| C|g(x)|, where x> k.The constants C and k are called the witnesses to the relationship. The definition of f(x) is O(g(x)) says that f(x) grows slower than some fixed multiple of g(x) as x grows without bound. Step-2: For our convenience let us consider , n is even. Consider , f(n) = n! We know that , n! = 1 × 2 × 3 × 4 × ....... × n = (1 × 3 × 5 × ..... × (n 1) × (2 × 4 × .... × n) = (1 × 3 × 5 × ..... × (n 1)...