When all the students in a school are made to stand in rows of $54,$ $30$ such rows are formed. If the students are made to stand in rows of $45,$ how many such rows will be formed?

Let $n(A) = 3$ and $n(B) = 3$ (where $n(S)$ denotes the number of elements in set $S$). Then,the number of subsets of $(A \times B)$ having an odd number of elements is:

$n$ balls each of weight $w$ are weighed in pairs. The sum of the weights of all possible pairs is $120$. When they are weighed in triplets,the sum of the weights of all possible triplets is $480$. Then $n$ is:

All possible numbers are formed using the digits $1, 1, 2, 2, 2, 2, 3, 4, 4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is

The number of natural numbers less than $7,000$ which can be formed by using the digits $0, 1, 3, 7, 9$ (repetition of digits allowed) is equal to

If in a regular polygon the number of diagonals is $54$,then the number of sides of this polygon is