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 $(7!)$)

The number of three-digit numbers in which $9$ appears in exactly one place is

If $^n{P_4} = 30 \times {^n}{C_5}$,then $n = $

If all the numbers which are greater than $6000$ and less than $10000$ are formed with the digits $3, 5, 6, 7, 8$ without repetition of the digits,then the difference between the number of odd numbers and the number of even numbers among them is

The students $S_{1}, S_{2}, \ldots, S_{10}$ are to be divided into $3$ groups $A, B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is ........ .

If $a + b + c = 50$ and $a, b, c$ are non-negative even integers,then the greatest value of $ab^2c$ is: