If $(n+1)! = 6[(n-1)!]$,find $n$.

The number of permutations of all the letters $AAAABBBC$ in which all the $A$'s appear together in a block of $4$ letters or all the $B$'s appear together in a block of $3$ letters,is-

$A$ question paper had ten questions. Each question could only be answered as True $(T)$ or False $(F)$. Each candidate answered all the questions. Yet,no two candidates wrote the answer in an identical sequence. How many different sequences of answers are possible?

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:

If $^{56}P_{r+6} : ^{54}P_{r+3} = 30800 : 1$,then $r = $

In an election,there are $5$ candidates and $3$ vacancies. $A$ voter can vote for a maximum of $3$ candidates. In how many ways can a voter cast their vote?