How many $5$-digit numbers can be formed using the digits $0, 1, 2, 3, 4,$ and $5$ without repetition such that the number is divisible by $3$?

If $^nP_3 + ^nC_{n-2} = 14n$,then $n = $

Let $n \geq 3$ be an integer. For a permutation $\sigma = (a_1, a_2, \ldots, a_n)$ of $(1, 2, \ldots, n)$,we define $f_\sigma(x) = a_n x^{n-1} + a_{n-1} x^{n-2} + \ldots + a_2 x + a_1$. Let $S_\sigma$ be the sum of the roots of $f_\sigma(x) = 0$ and let $S$ denote the sum over all permutations $\sigma$ of $(1, 2, \ldots, n)$ of the values $S_\sigma$. Then,

The number of three-digit numbers $\overline{abc}$ such that the arithmetic mean of $b$ and $c$ is equal to the square of their geometric mean 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:

In a hotel,four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is . . . . . . . .