Let $\mathbb{N}$ denote the set of all positive integers. Consider the sets

$$ A=\{1,2,3,4,5\} \text { and } B=\{1,2,3,4,5,6,7\} . $$

Let $S$ be the set of all functions $f: A \rightarrow B$ such that $f(2) \neq 2$ and $f(4) \neq 4$. Consider the set $T=\left\{f \in S:\right.$ there exists a function $g: B \rightarrow \mathbb{N}$ such that $g(f(x))=2^x$ for all $\left.x \in A\right\}$.

Then the number of elements in the set $T$ is $\_\_\_\_$ .

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $X$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $\alpha$ is the mean of the random variable $X$, then the value of $77 \alpha$ is $\_\_\_\_$ .

Consider the ellipse $E$ given by $\frac{x^2}{18}+\frac{y^2}{12}=1$. Let $H$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $E$ and whose foci are the same as that of $E$. Let $P$ and $Q$ be the points of intersection of $H$ and the parabola $\sqrt{5} y=x^2$ in the first quadrant. Let $d$ be the distance between $P$ and $Q$.

If $a$ and $b$ are the integers such that $d^2=a+b \sqrt{5}$, then the value of $a-b$ is $\_\_\_\_$ .