The number of ways of distributing $500$ dissimilar boxes equally among $50$ persons is:

$A$ question paper has two sections $A$ and $B$,in which section-$A$ has $8$ questions and section-$B$ has $6$ questions. $A$ student has to answer a total of $10$ questions,choosing at least $4$ questions from section-$A$ and at least $3$ questions from section-$B$. The number of ways a student can answer the paper is:

There are $7$ identical white balls and $3$ identical black balls. The number of distinguishable arrangements in a row of all the balls,so that no two black balls are adjacent is

Let $N$ be the set of positive integers. The number of distinct triplets $(x, y, z)$ satisfying $x, y, z \in N, x < y < z$ and $x+y+z=12$ is

$A$ linguistic club of a certain Institute consists of $6$ girls and $4$ boys. $A$ team of $4$ members is to be selected from this group,including the selection of a Captain (from among these $4$ members) for the team. If the team has to include at most one boy,the number of ways of selecting the team is:

The number of ways a committee of $8$ members can be formed from a group of $10$ men and $8$ women such that the committee contains at most $5$ men and at least $5$ women is: