The solution set of the system of equations $x + y = \frac{2\pi}{3}$ and $\cos x + \cos y = \frac{3}{2}$,where $x$ and $y$ are real,is:

If $S$ and $T$ are two sets such that $S$ has $21$ elements,$T$ has $32$ elements,and $S \cap T$ has $11$ elements,how many elements does $S \cup T$ have?

Let $A_1, A_2, \ldots, A_m$ be non-empty subsets of $\{1, 2, 3, \ldots, 100\}$ satisfying the following conditions:
$1.$ The numbers $|A_1|, |A_2|, \ldots, |A_m|$ are distinct.
$2.$ $A_1, A_2, \ldots, A_m$ are pairwise disjoint.
(Here $|A|$ denotes the number of elements in the set $A$).
Then,the maximum possible value of $m$ is:

Out of $800$ boys in a school,$224$ played cricket,$240$ played hockey,and $336$ played basketball. Of the total,$64$ played both basketball and hockey,$80$ played cricket and basketball,and $40$ played cricket and hockey. $24$ played all three games. The number of boys who did not play any game 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:

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 $C_{1}$ or chemical $C_{2}$.