$^{20}C_1 + 3 ^{20}C_2 + 3 ^{20}C_3 + ^{20}C_4$ is equal to-

Statement-$1$: $10$ identical balls can be distributed into $4$ distinct boxes in $^9C_3$ ways such that no box remains empty.
Statement-$2$: Any $3$ positions out of $9$ positions can be selected in $^9C_3$ ways.

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

To fill $12$ vacancies,there are $25$ candidates,of which $5$ are from the scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all,then the number of ways in which the selection can be made 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 ${}^9C_3$.

$A$ student is allowed to select at least $(n+1)$ books but not all books from a collection of $(2n+1)$ books. If the total number of ways in which he can select these books is $255$,then the number of books in that collection is