The probability of failing in Physics is $20\%$,and the probability of failing in Mathematics is $10\%$. What is the probability of failing in at least one subject (in $\%$)?

If $X$ and $Y$ are two sets such that $X$ has $40$ elements,$X \cup Y$ has $60$ elements and $X \cap Y$ has $10$ elements,how many elements does $Y$ have?

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 . . . . . .

Let $A = \{n \in N \mid n^{2} \leq n + 10,000\}$,$B = \{3k + 1 \mid k \in N\}$,and $C = \{2k \mid k \in N\}$. Then the sum of all the elements of the set $A \cap (B - C)$ is equal to $.....$

In a survey of $400$ students in a school,$100$ were listed as taking apple juice,$150$ as taking orange juice and $75$ were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.

In a class of $60$ students,$30$ opted for $NCC$,$32$ opted for $NSS$,and $24$ opted for both $NCC$ and $NSS$. If one of these students is selected at random,find the probability that the student opted for $NCC$ or $NSS$.