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

$A$ class has $175$ students. The following data shows the number of students obtaining one or more subjects: Mathematics $100$,Physics $70$,Chemistry $40$; Mathematics and Physics $30$,Mathematics and Chemistry $28$,Physics and Chemistry $23$; Mathematics,Physics and Chemistry $18$. How many students have offered Mathematics alone?

The domain of the function $f(x) = \frac{1}{1 + e^x}$ is

If $f(x) = \frac{x - |x|}{|x|}$,then $f(-1) = $

Let $A = \{1, 2\}, B = \{2, 3\}$. Evaluate $A \times B$.

If $A = \{x, y\}$,then the power set of $A$ is