Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of the first ten prime numbers. Let $A = S \cup P$,where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y)$,where $x \in S$ and $y \in A$,such that $x$ divides $y$,is . . . . . .

$A$ survey shows that $73 \%$ of the persons working in an office like coffee,whereas $65 \%$ like tea. If $x$ denotes the percentage of them who like both coffee and tea,then $x$ cannot be

In a city,$20\%$ of the population travels by car,$50\%$ travels by bus,and $10\%$ travels by both car and bus. Then,the percentage of persons travelling by car or bus is......$\%$

In a class of $100$ students,$15$ students chose only physics (but not mathematics and chemistry),$3$ chose only chemistry (but not mathematics and physics),and $45$ chose only mathematics (but not physics and chemistry). Of the remaining students,it is found that $23$ have taken physics and chemistry,$20$ have taken physics and mathematics,and $12$ have taken mathematics and chemistry. The number of students who chose all the three subjects is

$2n(A \setminus B) = n(B \setminus A)$ and $5n(A \cap B) = n(A) + 3n(B)$,where $P \setminus Q = P \cap Q^C$. If $n(A \cup B) \leq 10$,then the value of $\frac{n(A) \cdot n(B) \cdot n(A \cap B)}{8}$ is:

Consider the set $A$ of natural numbers $n$ whose units digit is non-zero,such that if this units digit is erased,then the resulting number divides $n$. If $K$ is the number of elements in the set $A$,then