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 $f(2) \neq 2$ and $f(4) \neq 4$. Consider the set $T = \{f \in S :$ there exists a function $g : B \to \mathbb{N}$ such that $g(f(x)) = 2^{x}$ for all $x \in A\}$. Then the number of elements in the set $T$ is ________.

$A = \{1,2,3,4,5\},\quad B = \{1,2,3,4,5,6,7\}$ $S = \{f: A\to B : f(2)\ne 2,\ f(4)\ne 4\}$ $T = \{f \in S : \exists\, g: B\to\mathbb{N},\ g(f(x)) = 2^x\ \forall x\in A\}$ $\text{To find: } |T|$ $1.\ T = \{\text{injective } f\}$ $2.\ \text{count injective } f: A\to B$ $3.\ \text{exclude } f(2)=2,\ f(4)=4$ $f(x_1) = f(x_2) \Rightarrow g(f(x_1)) = g(f(x_2))$ $\Rightarrow 2^{x_1} = 2^{x_2} \Rightarrow x_1 = x_2$ $f \text{ injective} \Rightarrow g(y) = 2^{f^{-1}(y)} \text{ on } f(A),\ \text{else } 1$ $g(f(x)) = 2^x\ \checkmark$ $T = \{f: A\to B \text{ injective},\ f(2)\ne 2,\ f(4)\ne 4\}$ $|\{f: A\to B \text{ injective}\}| = {}^{7}P_{5} = 7\cdot6\cdot5\cdot4\cdot3 = 2520$ $f(2)=2:\quad {}^{6}P_{4} = 6\cdot5\cdot4\cdot3 = 360$ $f(4)=4:\quad 360$ $f(2)=2 \text{ and } f(4)=4:\quad {}^{5}P_{3} = 5\cdot4\cdot3 = 60$ $|T| = 2520 - 360 - 360 + 60$ $= 1860$ $\boxed{|T| = 1860}$