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 . . . . . . .

There are $4$ notes of Rs. $100$ and $5$ other notes of denominations Rs. $1$,Rs. $2$,Rs. $5$,Rs. $20$,and Rs. $50$. These $9$ notes are to be distributed among $3$ children such that each child receives at least one note of Rs. $100$. Find the total number of ways of distribution.

Letters in the word $HULULULU$ are rearranged. The probability of all three $L$ being together is

If $^n{P_4} = 30 \times {^n}{C_5}$,then $n = $

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 ....

There are $4$ men and $5$ women in Group $A$,and $5$ men and $4$ women in Group $B$. If $4$ persons are selected from each group,then the number of ways of selecting $4$ men and $4$ women is: