Let $A, B, C$ be three non-void subsets of set $S$. Let $(A \cap C) \cup (B \cap C^{\prime}) = \phi$,where $C^{\prime}$ denotes the complement of set $C$ in $S$. Then:

$A$ number is chosen at random from the set $\{1, 2, 3, \ldots, 2000\}$. Let $p$ be the probability that the chosen number is a multiple of $3$ or a multiple of $7$. Then the value of $500p$ is . . . . . .

$A$ and $B$ are two sets having $3$ and $6$ elements respectively. Consider the following statements. Statement $(I)$: Minimum number of elements in $A \cup B$ is $6$. Statement $(II)$: Maximum number of elements in $A \cap B$ is $3$. Which of the following is correct?

Two newspapers $A$ and $B$ are published in a city. It is known that $25\%$ of the city population reads $A$ and $20\%$ reads $B$,while $8\%$ reads both $A$ and $B$. Further,$30\%$ of those who read $A$ but not $B$ look into advertisements,$40\%$ of those who read $B$ but not $A$ look into advertisements,and $50\%$ of those who read both $A$ and $B$ look into advertisements. The percentage of the population who look into advertisements is:

$A$ survey shows that $73 \%$ of the persons working in an office like coffee,whereas $65 \%$ like tea. If $x$ denotes the percentage of them who like both coffee and tea,then $x$ cannot be

Let $S = \{4, 6, 9\}$ and $T = \{9, 10, 11, \ldots, 1000\}$. If $A = \{a_{1} + a_{2} + \ldots + a_{k} : k \in N, a_{1}, a_{2}, \ldots, a_{k} \in S\}$,then the sum of all the elements in the set $T - A$ is equal to: