If all possible $4$-digit numbers are formed by choosing $4$ different digits from the given digits $1, 2, 3, 5, 8$,then the sum of all such $4$-digit numbers 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 . . . . . . . .

$A$ group of students comprises $5$ boys and $n$ girls. If the number of ways,in which a team of $3$ students can be randomly selected from this group such that there is at least one boy and at least one girl in each team,is $1750$,then $n$ is equal to

In the prime factorization of $37! = 2^{\alpha_2} \cdot 3^{\alpha_3} \cdot 5^{\alpha_5} \cdots 37^{\alpha_{37}}$,the ratio $\alpha_3 : \alpha_5$ is:

The number of $4$-digit numbers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7$ such that each number contains the digit $1$ is:

The number of seven-digit integers with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only is: