In a group of $3$ girls and $4$ boys,there are two boys $B_1$ and $B_2$. The number of ways in which these girls and boys can stand in a queue such that all the girls stand together,all the boys stand together,but $B_1$ and $B_2$ are not adjacent to each other,is:

For $n \geq 2$,let $S_n$ denote the set of all subsets of $\{1, 2, \ldots, n\}$ such that no two elements are consecutive. For example,$\{1, 3, 5\} \in S_6$,but $\{1, 2, 4\} \notin S_6$. Then $n(S_5)$ is equal to . . . . . . .

All the arrangements,with or without meaning,of the word $FARMER$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $FARMER$ in this list is .... .

Let $x, y, z$ be three non-negative integers such that $x+y+z=10$. The maximum possible value of $xyz+xy+yz+zx$ is

The number of positive integers less than $10000$ which contain the digit $5$ at least once is

If $^nC_r = ^nC_{r-1}$ and $^nP_r = ^nP_{r+1}$,then the value of $n$ is