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:

In a class of $55$ students,$23$ study Mathematics,$24$ study Physics,$19$ study Chemistry,$12$ study Mathematics and Physics,$9$ study Mathematics and Chemistry,$7$ study Physics and Chemistry,and $4$ study all three subjects. Find the number of students who study exactly one subject.

In a class of $100$ students,$55$ students have passed in Mathematics and $67$ students have passed in Physics. Then the number of students who have passed in Physics only is

If ${A_1}, {A_2}, {A_3}, \dots, {A_{30}}$ are $30$ sets each having $5$ elements and ${B_1}, {B_2}, \dots, {B_n}$ are $n$ sets each having $3$ elements. Let $\bigcup_{i=1}^{30} {A_i} = \bigcup_{j=1}^n {B_j} = S$ and each element of $S$ belongs to exactly $10$ of the $A_i$'s and exactly $9$ of the $B_j$'s,then $n$ is equal to:

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?

$A$ contractor has two teams of workers,team $A$ and team $B$. Team $A$ can complete a project $P$ in $12$ days and team $B$ can complete $P$ in $36$ days. Team $A$ starts working on $P$ and team $B$ joins team $A$ after four days. Team $A$ is withdrawn after another two days and team $B$ is asked to double its efficiency. The number of additional days required for team $B$ to complete $P$ is