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 $NSS$.

In a school of $800$ boys,$224$ play cricket,$240$ play hockey,and $336$ play basketball. Of the total,$64$ play basketball and hockey,$80$ play cricket and basketball,and $40$ play cricket and hockey,while $24$ play all three games. Find the number of boys who do not play any game.

The probabilities that a student passes in Mathematics,Physics,and Chemistry are $m, p$,and $c$ respectively. The student has a $75\%$ chance of passing in at least one,a $50\%$ chance of passing in at least two,and a $40\%$ chance of passing in exactly two. Which of the following relations are true?

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $1, 6, 11, \dots$ and $Y$ be the set consisting of the first $2018$ terms of the arithmetic progression $9, 16, 23, \dots$. Then,the number of elements in the set $X \cup Y$ is:

$A$ survey shows that $63 \%$ of the people in a city read newspaper $A$ whereas $76 \%$ read newspaper $B$. If $x \%$ of the people read both the newspapers,then a possible value of $x$ can be

Let $S = \{1, 2, 3, 5, 7, 10, 11\}$. The number of nonempty subsets of $S$ such that the sum of all elements is a multiple of $3$ is $........$