The least value of $n$ such that ${}^{(n-1)}C_6 + {}^{(n-1)}C_7 < {}^{n}C_8$ is

In an election,$3$ members are to be elected from $6$ candidates. $A$ voter can cast their vote for any number of candidates but not more than the number of candidates to be elected. In how many ways can a voter cast their vote?

In how many ways can $6$ '$+$' signs and $4$ '$*$' signs be arranged in a row such that no two '$*$' signs occur together?

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are all $4$ cards face cards?

$A$ student is asked to answer $10$ out of $13$ questions in an examination such that he must answer at least four questions from the first five questions. Then the total number of possible choices available to him is