There are $6$ candidates for $3$ posts. In how many ways can the posts be filled?

The number of ways in which five identical balls can be distributed among ten identical boxes such that no box contains more than one ball,is

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

$3$ boys and $3$ girls are to be seated around a circular table. Among the boys,$X$ does not want any girl as a neighbour,and among the girls,$Y$ does not want any boy as a neighbour. How many such arrangements are possible?

In a circus, there are $10$ cages to accommodate $10$ animals. Out of these, $4$ cages are so small that $5$ specific animals cannot enter them. In how many ways is it possible to accommodate the $10$ animals in these $10$ cages?

In how many different ways can the letters of the word '$CORPORATION$' be arranged so that the vowels always come together?