The number of $3$-digit numbers that are divisible by either $2$ or $3$ but not divisible by $7$ is $.........$.

In a class of $300$ students,every student reads $5$ newspapers and every newspaper is read by $60$ students. Then the number of newspapers 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

Let $S$ be the set of the first $11$ natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \ge 2 \text{ and the product of all elements of } B \text{ is even}\}$ is . . . . . . .

If $X$ and $Y$ are two sets such that $n(X) = 17$,$n(Y) = 23$ and $n(X \cup Y) = 38$,find $n(X \cap Y)$.

The number of $3$-digit numbers that are divisible by either $3$ or $4$ but not divisible by $48$ is: