$60$ words can be made using all the letters of the word $BHBJO$,with or without meaning. If these words are written as in a dictionary,then the $50^{\text{th}}$ word is:

The total number of permutations of $n (> 1)$ different things taken not more than $r$ at a time,when each thing may be repeated any number of times is

The number of different nine-digit numbers that can be formed by rearranging the digits of the number $223355888$ such that the odd digits occupy even positions is:

The domain of the function $f(x) = {}^{7-x}P_{x-1}$ is:

$A$ library has $a$ copies of one book,$b$ copies of each of two books,$c$ copies of each of three books,and single copies of $d$ books. The total number of ways in which these books can be arranged is:

For $n = 1, 2, 3, . . . , 50$,let $A = \{ a_n \mid a_n = \begin{cases} (-1)^{\frac{n}{2}} (\frac{n}{2}), & \text{if } n \text{ is even} \\ (-1)^{\frac{n-1}{2}} (\frac{n-1}{2}), & \text{if } n \text{ is odd} \end{cases} \}$ and $B$ be the set of all distinct elements of $A$. The number of permutations of all the elements of set $B$ such that even integers are in increasing order is: