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

There are $15$ players in a cricket team,out of which $6$ are bowlers,$7$ are batsmen,and $2$ are wicketkeepers. The number of ways a team of $11$ players can be selected from them so as to include at least $4$ bowlers,$5$ batsmen,and $1$ wicketkeeper is $.....$

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

The number of $4$-digit numbers without repetition that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ in which each number has two odd digits and two even digits is:

$A$ student has to answer $10$ out of $13$ questions in an examination,choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student is:

Consider three boxes,each containing $10$ balls labelled $1, 2, \dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is: