If ${ }^{n} C_{12}={ }^{n} C_{8}$,then $n$ is equal to

Number of ways of arranging $8$ identical books into $4$ identical shelves where any number of shelves may remain empty is equal to

$A$ committee of $7$ members is to be formed from $9$ boys and $4$ girls. In how many ways can this be done if the committee must consist of exactly $3$ girls?

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

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

$A$ student has to answer $10$ out of $13$ questions in an examination. The number of ways in which he can answer if he must answer at least $3$ of the first $5$ questions is: