We wish to select $6$ persons from $8,$ but if the person $A$ is chosen,then $B$ must be chosen too. In how many ways can the selection be made?

$A$ 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:

Out of $5$ apples,$10$ mangoes,and $15$ oranges,$15$ fruits are to be distributed between two persons. Find the total number of ways of distribution.

Suppose that $20$ pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar is connected by beams to the top of all its non-adjacent pillars,then the total number of beams is:

$n$-digit numbers are formed using only three digits $2, 5$,and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed is:

How many $3$-digit numbers are there,with no digits repeated?