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

In a Mathematics examination,there are $20$ questions of equal marks. The question paper is divided into three sections: $A, B$,and $C$. $A$ student is required to attempt a total of $15$ questions,taking at least $4$ questions from each section. If section $A$ has $8$ questions,section $B$ has $6$ questions,and section $C$ has $6$ questions,then the total number of ways a student can select $15$ questions is:

If ${}^{n-1}C_3 + {}^{n-1}C_4 > {}^{n}C_3$, then $n$ is just greater than which integer?

$A$ bag contains $5$ black and $6$ red balls. Determine the number of ways in which $2$ black and $3$ red balls can be selected.

The least value of $n$ such that ${ }^{(n-1)} C_3 + { }^{(n-1)} C_4 > { }^n C_3$ is:

Find the number of ways of selecting $9$ balls from $6$ red balls,$5$ white balls,and $5$ blue balls if each selection consists of $3$ balls of each colour.