The number of positive integral solutions of the equation $xyz = 3000$ is

Consider the following statements:
$i.$ The number of ways of placing $n$ distinct objects in $k$ distinct bins $(k \leq n)$ such that no bin is empty is ${}^{n-1}C_{k-1}$.
$ii.$ The number of ways of writing a positive integer $n$ as a sum of $k$ positive integers is ${}^{n-1}C_{k-1}$.
$iii.$ The number of ways of placing $n$ distinct objects in $k$ distinct bins such that at least one bin is non-empty is ${}^{n-1}C_{k-1}$.
$iv.$ ${}^nC_k - {}^{n-1}C_k = {}^{n-1}C_{k-1}$.

There are $10$ pairs of shoes in a cupboard from which $4$ shoes are picked at random. The probability that there is at least one pair is:

The number of ways $5$ boys and $4$ girls can sit in a row so that either all the boys sit together or no two boys sit together,is $......$

If the number of all possible permutations of the letters of the word $MATHEMATICS$ in which the repeated letters are not together is $90(X)$,then $X=$

For a set of $5$ true or false questions,no student has written all the correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is