In a class of $30$ pupils,$12$ take needle work,$16$ take physics,and $18$ take history. If all the $30$ students take at least one subject and no one takes all three,then the number of pupils taking exactly $2$ subjects is:

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 . . . . . .

In a class of $60$ students, $40$ opted for $NCC$, $30$ opted for $NSS$, and $20$ opted for both $NCC$ and $NSS$. If one of these students is selected at random, then the probability that the selected student has opted for neither $NCC$ nor $NSS$ 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

Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers satisfying the condition $x^2 - y^2 = 12345678$. Then,

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ and $B = \{3, 6, 7, 9\}$. Then the number of elements in the set $\{ C \subseteq A : C \cap B \neq \phi \}$ is