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

From $6$ different novels and $3$ different dictionaries,$4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

The number of times the digit $5$ will be written when listing the integers from $1$ to $1000$ is

Let $m$ (respectively,$n$) be the number of $5$-digit integers obtained by using the digits $1, 2, 3, 4, 5$ with repetitions (respectively,without repetitions) such that the sum of any two adjacent digits is odd. Then $\frac{m}{n}$ is equal to

The sum of all $4$-digit numbers that can be formed using the digits $2, 3, 4, 5, 6$ without repetition is:

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)