If sets $A$ and $B$ have $3$ and $6$ elements respectively,what is the minimum number of elements in $A \cup B$?

If $A = \{1, 2, 3, 4, 5, 6\}$,then the number of subsets of $A$ which contain at least two elements is

Let $E$ denote the set of letters of the English alphabet,$V = \{a, e, i, o, u\}$ and $C$ be the complement of $V$ in $E$. Then,the number of four-letter words (where repetitions of letters are allowed) having at least one letter from $V$ and at least one letter from $C$ is

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ and $B = \{3, 6, 7, 9\}$. Then the number of elements in the set $\{ C \subseteq A : C \cap B \neq \phi \}$ is

In a group of students,$100$ students know Hindi,$50$ know English,and $25$ know both. Each student knows either Hindi or English. How many students are there in the group?

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: