If a seven-digit number formed with distinct digits $4, 6, 9, 5, 3, x,$ and $y$ is divisible by $3$,then the number of such ordered pairs $(x, y)$ is

Let $S$ denote the set of $4$-digit numbers $abcd$ such that $a > b > c > d$ and $P$ denote the set of $5$-digit numbers having the product of its digits equal to $20$. Then $n(S) + n(P)$ is equal to:

The number of distinct primes dividing $12! + 13! + 14!$ is

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets such that they get consecutive blocks of $5$,$3$,and $2$ tickets is:

The number of $3$-digit odd numbers divisible by $3$ that can be formed using the digits $1, 2, 3, 4, 5, 6$ when repetition is not allowed is:

$A$ building has a ground floor and $10$ more floors. Nine persons enter a lift at the ground floor. The lift goes up to the $10$th floor. The number of ways in which any $4$ persons exit at a floor and the remaining $5$ persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to: