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?

In how many ways can $12$ balls be distributed between two friends such that one gets $8$ balls and the other gets $4$ balls?

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men such that it contains at least $5$ women. Find the number of ways the committee can be formed such that women are in the majority and the number of ways the committee can be formed such that men are in the majority,respectively.

The number of ways in which a team of $11$ players can be formed out of $25$ players,if $6$ out of them are always to be included and $5$ of them are always to be excluded,is

$A$ question paper has two sections $A$ and $B$,in which section-$A$ has $8$ questions and section-$B$ has $6$ questions. $A$ student has to answer a total of $10$ questions,choosing at least $4$ questions from section-$A$ and at least $3$ questions from section-$B$. The number of ways a student can answer the paper is:

If $\binom{n-1}{r} = (k^2 - 3) \binom{n}{r+1}$,then $k \in \dots$