If $^n{P_r} = 840$ and $^n{C_r} = 35$,then $n$ is equal to

The difference between an integer $n$ and its cube $n^3$ is always divisible by which of the following?

If $_nP_r = 30240$ and $\binom{n}{r} = 252$,then $(n, r) = \dots$

If a five-digit number divisible by $3$ is to be formed using the digits $0, 1, 2, 3, 4,$ and $5$ without repetition,then the total number of ways this can be done is:

In a hotel,four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is . . . . . . . .

Let $x$ denote the number of ways of selecting at least one ball from a bag containing $3$ identical red balls,$4$ identical blue balls,and $5$ identical green balls. Let $y$ denote the number of ways in which a student can fail in an examination,when they have to write the examination in $5$ different subjects. Then $x+y=$