Three schools send $2, 4,$ and $6$ students,respectively,to a summer camp. The $12$ students must be accommodated in $6$ rooms numbered $1, 2, 3, 4, 5, 6$ in such a way that each room has exactly $2$ students and both are from the same school. The number of ways the students can be accommodated in the rooms is

The number of integers lying between $1000$ and $10000$ such that every number contains the digits $3$ and $7$ exactly once without repetition is:

If $_nP_r = 30240$ and $\binom{n}{r} = 252$,then $(n, r) = \dots$

$A$ letter lock consists of three rings with $15$ different letters each. If $N$ denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then:

Five boys and five girls are writing an examination in a hall in which $5$ benches are arranged in a row and only two students are to be seated on every bench at either of its ends. If the seating arrangement is to be such that no two boys or no two girls sit together as neighbours (a student should not have a student of same gender either on left or right),then the total number of such arrangements is

Consider the set $A = \{1, 2, 3, \ldots, 30\}$. The number of ways in which one can choose three distinct numbers from $A$ such that the product of the chosen numbers is divisible by $9$ is: