In how many ways can $15$ different books be divided equally among $3$ sets or groups?

Number of shortest paths from $HOSTEL$ to $ALLEN$ is equal to (as shown in the given figure):

$A$ man $X$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. The total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men,such that $3$ friends of each of $X$ and $Y$ are in this party,is:

How many lines can be drawn through $21$ points on a circle?

$A$ committee of $11$ members is to be formed from $8$ men and $5$ women. If $m$ is the number of ways to form the committee with at least $6$ men and $n$ is the number of ways to form the committee with at least $3$ women,then:

In how many different ways can the letters of the word '$AUCTION$' be arranged in such a way that the vowels always come together?