The sum of all the elements in the set $\{n \in \{1, 2, \ldots, 100\} \mid \text{H.C.F. of } n \text{ and } 2040 \text{ is } 1\}$ is equal to $.....$

If $A$ and $B$ are events of a random experiment such that $P(A \cup B) = \frac{3}{4}$,$P(A \cap B) = \frac{1}{4}$,and $P(\bar{A}) = \frac{2}{3}$,then find $P(\bar{A} \cap B)$.

In a school,$20$ teachers teach either Mathematics or Physics. If $12$ teachers teach Mathematics and $4$ teachers teach both subjects,then the number of teachers teaching only Physics is:

Let $X$ be a set of $5$ elements. The number $d$ of ordered pairs $(A, B)$ of subsets of $X$ such that $A \neq \phi, B \neq \phi, A \cap B \neq \phi$ satisfies:

$205$ students take an examination,of whom $105$ pass in English,$70$ students pass in Mathematics,and $30$ students pass in both. How many students fail in both subjects?

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: