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:

In a class of $60$ students,$30$ opted for $NCC$,$32$ opted for $NSS$,and $24$ opted for both $NCC$ and $NSS$. If one of these students is selected at random,find the probability that the student opted for $NCC$ or $NSS$.

$A$ contractor has two teams of workers,team $A$ and team $B$. Team $A$ can complete a project $P$ in $12$ days and team $B$ can complete $P$ in $36$ days. Team $A$ starts working on $P$ and team $B$ joins team $A$ after four days. Team $A$ is withdrawn after another two days and team $B$ is asked to double its efficiency. The number of additional days required for team $B$ to complete $P$ is

The number of $3$-digit numbers that are divisible by either $3$ or $4$ but not divisible by $48$ is:

The sum of the integers from $1$ to $100$ which are not divisible by $3$ or $5$ is

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