$A$ student is allowed to select at least $(n+1)$ books but not all books from a collection of $(2n+1)$ books. If the total number of ways in which he can select these books is $255$,then the number of books in that collection is

If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

If $^nC_4, ^nC_5,$ and $^nC_6$ are in $A.P.,$ then $n$ can be

From a group of $10$ men and $8$ women,the number of ways of forming a committee of $8$ members with not more than $5$ men and not less than $5$ women is

The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples,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: