Let $S = \{(a, b) : a, b \in \mathbb{Z}, 0 \leq a, b \leq 18\}$. The number of elements $(x, y)$ in $S$ such that $3x + 4y + 5$ is divisible by $19$ is:

If $P(A) = 0.5$,$P(B) = 0.7$,and $P(A \cap B) = 0.6$,then $P(A \cup B) = \dots$

There are $100$ students in a class. In an examination, $50$ of them failed in Mathematics, $45$ failed in Physics, $40$ failed in Biology and $32$ failed in exactly two of the three subjects. Only one student passed in all the subjects. Then, the number of students failing in all the three subjects is:

$X, Y, Z$ are the sets of all positive divisors of $10^{60}, 20^{50},$ and $30^{40}$ respectively. Find $n(X \cup Y \cup Z)$.

In a group of $400$ people,$250$ can speak Hindi and $200$ can speak English. How many people can speak both Hindi and English?

Let $A = \{1, 6, 11, 16, \dots\}$ and $B = \{9, 16, 23, 30, \dots\}$ be the sets consisting of the first $2025$ terms of two arithmetic progressions. Then $n(A \cup B)$ is