In how many ways can $3n$ distinct objects be distributed equally among $3$ persons?

The number of different words that can be formed using all the letters of the word $APPLICATION$ such that no two vowels come together is -

Find the number of natural number solutions to the equation $x_1 + x_2 = 100$,such that $x_1$ and $x_2$ are not multiples of $5$.

The students $S_{1}, S_{2}, \ldots, S_{10}$ are to be divided into $3$ groups $A, B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is ........ .

The total number of selections of the letters in the phrase "ned needs nineteen nets" is:

The total number of positive integral solutions $(x, y, z)$ such that $xyz = 24$ is