There are $6$ different novels and $3$ different poetry books on a table. If $4$ novels and $1$ poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle,then the number of such possible arrangements is

The number of groups that can be made from $5$ different green balls,$4$ different blue balls,and $3$ different red balls,if at least $1$ green and $1$ blue ball must be included.

How many words of $4$ distinct letters can be formed using the letters of the word $DHOLPUR$ such that $L$ and $P$ are always included?

Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples,$5$ white apples,and $8$ oranges. If in the selected $5$ fruits,at least $2$ oranges,at least $1$ red apple,and at least $1$ white apple must be included,then the number of ways Anil's mother can offer $5$ fruits to Anil is $........$

If $(2 \leq r \leq n)$, then ${}^{n}C_{r} + 2 \cdot {}^{n}C_{r+1} + {}^{n}C_{r+2}$ is equal to:

If $^{n-1}C_r = (k^2 - 3) \cdot ^nC_{r+1}$,then the range of $k$ is: