$A$ student is required to answer $10$ questions out of $13$ in an examination,such that he must choose at least $4$ questions from the first $5$ questions. How many ways can he make the selection?

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?

In an election,the number of candidates is $1$ more than the number of persons to be elected. If the voters can cast their votes in $254$ ways,find the number of candidates. ($A$ voter cannot cast more votes than the number of persons to be elected.)

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

In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation?

$A$ set contains $(2n + 1)$ elements. The number of subsets of the set which contain at most $n$ elements is: