If $12 \cdot {}^{n}C_{2} = {}^{2n}C_{3}$,find $n$.

The number of ordered pairs $(r, k)$ for which $6 \cdot ^{35} C_{r} = (k^2 - 3) \cdot ^{36} C_{r+1}$,where $k$ is an integer,is:

In how many ways can $mn$ letters be posted in $n$ letter-boxes?

Each section in the first year of a plus $2$ course has exactly $40$ students. If there are $5$ sections,in how many ways can a set of $4$ student representatives be selected,$1$ from each section?

The total number of ways of selecting six coins out of $20$ one-rupee coins,$10$ fifty-paise coins,and $7$ twenty-five-paise coins is:

In how many ways can $5$ speakers $S_1, S_2, S_3, S_4$,and $S_5$ give speeches one after the other if $S_3$ wants to speak after $S_1$ and $S_2$?