Question

Let \(\mathbb{N}\) denote the set of all positive integers. Consider the sets \[ A = \{1,2,3,4,5\} \] and \[ B = \{1,2,3,4,5,6,7\}. \] Let \(S\) be the set of all functions \[ f : A \to B \] such that \[ f(2) \ne 2 \quad \text{and} \quad f(4) \ne 4. \] Consider the set \[ T = \left\{ f \in S : \text{there exists a function } g : B \to \mathbb{N} \text{ such that } g(f(x)) = 2^x \text{ for all } x \in A \right\}. \] Then the number of elements in the set \(T\) is:

Hint:

Whenever a problem defines a composite function $g(f(x)) = h(x)$ where $h(x)$ is injective, $f(x)$ must also be injective. This simplifies function-counting problems into permutation problems ($^nP_r$).

Answer 1

Correct Answer: 1860