The number of ways in which $4$ different things can be distributed to $6$ persons so that no person gets all the things is

The number of $3$-$digit$ odd numbers,whose sum of digits is a multiple of $7$,is

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

There are three sections in a question paper,each containing $4$ questions. If a candidate has to answer only $5$ questions from this paper without leaving any section,then the number of ways the candidate can make the choice of questions is:

The total number of ways in which $5$ balls of different colours can be distributed among $3$ persons so that each person gets at least one ball is

If $f(n) = n! (31-n)!$,where $n \in \{0, 1, 2, \ldots, 31\}$,then the minimum value of $f(n)$ is