Let $A = \{1, 6, 11, 16, \dots\}$ and $B = \{9, 16, 23, 30, \dots\}$ be the sets consisting of the first $2025$ terms of two arithmetic progressions. Then $n(A \cup B)$ is

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.

Let $X = \{n \in N : 1 \leq n \leq 50\}$. If $A = \{n \in X : n \text{ is a multiple of } 2\}$ and $B = \{n \in X : n \text{ is a multiple of } 7\}$,then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is

The number of $4$-digit numbers which are neither a multiple of $7$ nor a multiple of $3$ is ..... .

If the probabilities of a student succeeding in the entrance tests for institutes $A$, $B$ and $C$ are $0.6$, $0.5$ and $0.4$ respectively, while the probability of succeeding in both $A$ and $B$ is $0.3$, in both $B$ and $C$ is $0.2$, in both $A$ and $C$ is $0.2$, and in all three is $0.1$, then the probability that the student succeeds in exactly one of these tests is......

In a hostel,$60 \%$ of the students read Hindi newspaper,$40 \%$ read English newspaper and $20 \%$ read both Hindi and English newspapers. $A$ student is selected at random. Find the probability that she reads neither Hindi nor English newspapers.