The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ is

$\binom{n}{r+1} + 2\binom{n}{r} + \binom{n}{r-1} = \dots$

The number of ways in which $17$ apples can be distributed among $4$ guests such that each guest gets at least $3$ apples is

$A$ committee of $4$ persons is to be formed from $2$ ladies,$2$ old men,and $4$ young men such that it includes at least $1$ lady,at least $1$ old man,and at most $2$ young men. The total number of ways in which this committee can be formed is:

$^{n-1}C_r = (k^2 - 8) ^nC_{r+1}$ if and only if:

$A$ 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