If ${ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3$,then the minimum value of $n$ is

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

The number of non-negative integral solutions of the equation $x+y+z+t=10$ when $x \geq 2$ and $z \geq 5$ is:

The total number of all those $3$-digit numbers in which the sum of all the digits in each of them is $10$,is

$A$ question paper is divided into two parts,$A$ and $B$,each consisting of $5$ questions. If a candidate has to select $6$ questions in total,in how many ways can they do so if they must select at least $2$ questions from each part?

If $\frac{{}^{n+1}C_{r+1}}{{}^{n+1}C_r} = \frac{n-r+1}{m}$,then $m =$