If the $LCM$ of $p$ and $q$ is $r^2t^4s^2$,where $r, s,$ and $t$ are prime numbers,and $p$ and $q$ are positive integers,then the number of ordered pairs $(p, q)$ is ....

$n$ objects are distributed at random among $n$ persons. The number of ways in which this can be done so that at least one of them will not get any object is

If the number of circular permutations of $9$ distinct things taken $5$ at a time is $n_1$ and the number of linear permutations of $8$ distinct things taken $4$ at a time is $n_2$,then $\frac{n_1}{n_2}=$

There are $n$ white and $n$ black balls marked $1, 2, 3, \ldots, n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours is

The number of natural numbers less than $10000$ which are divisible by $5$ and in which no digit is repeated is:

$A$ question paper contains $4$ questions,each having $4$ alternative answers. The number of ways that a candidate can answer one or more questions is