$^nC_{r+1} + ^nC_{r-1} + 2^nC_r = \dots$

If $1 \times 1! + 2 \times 2! + 3 \times 3! + \ldots + n \times n! = 11! - 1$,then the maximum value of ${}^n C_r$ is

Out of $13$ cricket players,$4$ are bowlers. In how many ways can a team of $11$ players be selected such that the team contains at least $2$ bowlers?

Let $S_r = \{(x, y, z) : x + y + z = 11, x \geq r, y \geq r, z \geq r, x, y, z, r \in \mathbb{Z}\}$ and $n(S_r)$ represents the number of elements in $S_r$. Then $n(S_2) + n(S_3) + n(S_4) = $

If a committee consists of $3$ men and $2$ women,in how many ways can a committee of $5$ members be formed from $5$ men and $4$ women?

If $20$ identical books are to be distributed among $4$ persons such that each person receives at least one book,in how many ways can this be done?