Let $m = (9n^2 + 54n + 80)(9n^2 + 45n + 54)(9n^2 + 36n + 35)$. The greatest positive integer which divides $m$ for all positive integers $n$ 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:

The total number of integral solutions of the equation $xyz = 24$ is

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets among themselves such that they get consecutive blocks of $5, 3$ and $2$ tickets is

If ${}^n C_{r-1}=36$,${}^n C_r=84$,and ${}^n C_{r+1}=126$,then the value of ${}^n C_8$ is

Five balls of different colors are to be placed in three boxes of different sizes,where each box can hold all five balls. In how many ways can the balls be placed such that no box remains empty?