If $P(A) = \frac{2}{5}$,$P(B) = \frac{1}{4}$ and $P(A \cup B) = \frac{1}{2}$,then $P(A' \cup B') = $

The sum of all integers from $1$ to $100$ that are divisible by $2$ or $5$ is...........

There are $200$ individuals with a skin disorder,$120$ had been exposed to the chemical $C_{1}$,$50$ to chemical $C_{2}$,and $30$ to both the chemicals $C_{1}$ and $C_{2}$. Find the number of individuals exposed to chemical $C_{2}$ but not chemical $C_{1}$.

Each set $X_r$ contains $5$ elements and each set $Y_r$ contains $4$ elements. If $\bigcup_{r=1}^{24} X_r = S = \bigcup_{r=1}^{n} Y_r$,and each element of set $S$ belongs to exactly $10$ of the $X_r$'s and to exactly $6$ of the $Y_r$'s,then find the value of $n$.

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:

In a school,there are $20$ teachers who teach mathematics or physics. Of these,$12$ teach mathematics and $4$ teach both physics and mathematics. How many teach physics?