$A$ 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 $S$ be the set of all ordered pairs $(x, y)$ of positive integers,with $\text{HCF}(x, y) = 16$ and $\text{LCM}(x, y) = 48000$. The number of elements in $S$ is

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

Let $A = \{ x \in R : [x + 3] + [x + 4] \leq 3 \}$ and $B = \{ x \in R : 3^x \left( \sum_{n=1}^{\infty} \frac{3}{10^n} \right)^{x-3} < 3^{-3x} \}$,where $[t]$ denotes the greatest integer function. Then,

Let $A$ be the set of first $101$ terms of an $A$.$P$.,whose first term is $1$ and the common difference is $5$, and let $B$ be the set of first $71$ terms of an $A$.$P$.,whose first term is $9$ and the common difference is $7$. Then, the number of elements in $A \cap B$ which are divisible by $3$ is:

For a real number $x$,$[x]$ denotes the greatest integer less than or equal to $x$. The value of $\left[ \frac{1}{2} \right] + \left[ \frac{1}{2} + \frac{1}{100} \right] + \left[ \frac{1}{2} + \frac{2}{100} \right] + \dots + \left[ \frac{1}{2} + \frac{99}{100} \right]$ is