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

Out of $6$ boys and $4$ girls,a group of $7$ is to be formed. In how many ways can this be done if the group is to have a majority of boys?

The product of $r$ consecutive natural numbers is always divisible by......

$A$ group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least $3$ girls?

The least value of $n$ for which ${ }^{(n-1)} C_2 + { }^{(n-1)} C_3 > { }^n C_2$ is

In an examination,a question paper consists of $12$ questions divided into two parts,i.e.,Part $I$ and Part $II$,containing $5$ and $7$ questions,respectively. $A$ student is required to attempt $8$ questions in all,selecting at least $3$ from each part. In how many ways can a student select the questions?