In how many ways can a group of $7$ persons be formed from $6$ boys and $4$ girls such that the boys are in the majority?

Out of $14$ cricket players,$5$ are bowlers. In how many ways can a team of $11$ players be selected such that the team contains at least $4$ bowlers?

The least value of the natural number $n$ satisfying $C(n, 5)+C(n, 6)>C(n+1, 5)$ is

If $\binom{n-1}{4}$,$\binom{n-1}{5}$,and $\binom{n-1}{6}$ are in arithmetic progression,find $n$.

The greatest positive integer which divides $(n+16)(n+17)(n+18)(n+19)$,for all positive integers $n$,is

Statement $I$: The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is ${}^9C_3$.
Statement $II$: The number of ways of choosing $3$ places from $9$ different places is ${}^9C_3$.