Let $S$ be the set of the first $11$ natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \ge 2 \text{ and the product of all elements of } B \text{ is even}\}$ is . . . . . . .

There is a group of $265$ persons who like either singing, dancing, or painting. In this group, $200$ like singing, $110$ like dancing, and $55$ like painting. If $60$ persons like both singing and dancing, $30$ like both singing and painting, and $10$ like all three activities, then the number of persons who like only dancing and painting is:

Out of $60 \%$ female and $40 \%$ male candidates appearing in an exam,$60 \%$ of the total candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. $A$ candidate is randomly chosen from the qualified candidates. The probability that the chosen candidate is a female is:

The number of $3$-digit numbers that are divisible by either $3$ or $4$ but not divisible by $48$ is:

If $U$ is the universal set with $100$ elements; $A$ and $B$ are two sets such that $n(A)=50$,$n(B)=60$,and $n(A \cap B)=20$,then find $n(A^{\prime} \cap B^{\prime})$.

In a battle,$70\%$ of the soldiers lost one eye,$80\%$ lost one ear,$75\%$ lost one arm,and $85\%$ lost one leg. If $x\%$ of the soldiers lost all four,what is the minimum value of $x$?