The number of polynomials $p(x)$ with integer coefficients such that the curve $y=p(x)$ passes through $(2,2)$ and $(4,5)$ 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

If $A=\{x \in R: \sqrt{x^2-8x+15} \in R\}$ and $B=\{x \in R: \frac{x-3}{2x-5} < \frac{x-6}{2x-11}\}$, then $A \cap B=$

If $A, B$ and $C$ are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$,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:

Let $S = \{1, 2, 3, 4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to