Question 10

To satisfy the condition $g(f(x)) = 2^x$ for all $x \in A$, the function $f$ must be injective. If $f(x_1) = f(x_2)$, then $g(f(x_1)) = g(f(x_2)) \Rightarrow 2^{x_1} = 2^{x_2} \Rightarrow x_1 = x_2$. Thus, $T$ is the set of all injective functions $f: A \to B$ such that $f(2) \neq 2$ and $f(4) \neq 4$.

1. Total number of injective functions $f: A \to B$ is $^7P_5 = 7 \times 6 \times 5 \times 4 \times 3 = 2520$.
2. Let $P_1$ be the condition that $f(2) = 2$. Number of injective functions with $f(2) = 2$ is $^6P_4 = 6 \times 5 \times 4 \times 3 = 360$.
3. Let $P_2$ be the condition that $f(4) = 4$. Number of injective functions with $f(4) = 4$ is $^6P_4 = 360$.
4. Number of injective functions where both $f(2) = 2$ and $f(4) = 4$ is $^5P_3 = 5 \times 4 \times 3 = 60$.

Using the principle of inclusion-exclusion, the number of injective functions where $f(2) = 2$ or $f(4) = 4$ is $360 + 360 - 60 = 660$. The number of injective functions in $S$ is $2520 - 660 = 1860$. Final Answer: 1860.