Out of $8$ students in a classroom,$4$ of them are chosen and they are arranged around a table. If the remaining $4$ are arranged in a row,then the total number of arrangements that can be made with those $8$ students is

Let $P_m$ stand for $^nP_m$. Then the expression $1 \cdot P_1 + 2 \cdot P_2 + 3 \cdot P_3 + \dots + n \cdot P_n =$

Group $A$ consists of $7$ boys and $3$ girls,while group $B$ consists of $6$ boys and $5$ girls. The number of ways,$4$ boys and $4$ girls can be invited for a picnic if $5$ of them must be from group $A$ and the remaining $3$ from group $B$,is equal to:

If the number of seven-digit numbers,such that the sum of their digits is even,is $m \cdot n \cdot 10^{n}$; $m, n \in \{1, 2, 3, \ldots, 9\}$,then $m+n$ is equal to . . . . . . .

There are three sections in a question paper,each section containing $4$ questions. If a candidate has to answer exactly $5$ questions from this paper such that at least one question is answered from each section,then the number of ways in which a candidate can make the choice of questions is

The number of $5$-digit numbers that are not divisible by $5$ and consist of different odd digits is: