Let $A$ and $B$ be finite sets and $P_A$ and $P_B$ respectively denote their power sets. If $P_B$ has $112$ elements more than those in $P_A$,then the number of injective functions from $A$ to $B$ is

$A$ two-digit number $\overline{ab}$ is called almost prime if one obtains a two-digit prime number by changing at most one of its digits $a$ or $b$. (For example,$18$ is an almost prime number because $13$ is a prime number). The number of almost prime two-digit numbers is:

Let $A$ denote the set of all real numbers $x$ such that $x^3-[x]^3=(x-[x])^3$,where $[x]$ is the greatest integer less than or equal to $x$. Then,

If $P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ is a polynomial such that $P(0) = 1, P(1) = 2, P(2) = 5, P(3) = 10$ and $P(4) = 17$,then $P(5) =$

Let $x_k$ be real numbers such that $x_k \geq k^4+k^2+1$ for $1 \leq k \leq 2018$. Denote $N=\sum_{k=1}^{2018} k$. Consider the following inequalities.
$I$. $\left(\sum_{k=1}^{2018} k x_k\right)^2 \leq N\left(\sum_{k=1}^{2018} k x_k^2\right)$
$II$. $\left(\sum_{k=1}^{2018} k x_k\right)^2 \leq N\left(\sum_{k=1}^{2018} k^2 x_k^2\right)$
Then,

Let $S = \{(m, n): m, n \in \{1, 2, 3, \ldots, 50\}\}$. If the number of elements $(m, n)$ in $S$ such that $6^{m} + 9^{n}$ is a multiple of $5$ is $p$ and the number of elements $(m, n)$ in $S$ such that $m + n$ is a square of a prime number is $q$, then $p + q$ is equal to :