If all the letters of the word $ACADEMICIAN$ are permuted in all possible ways,then the number of permutations in which no two $A$'s are together and all the consonants are together is:

If $x$ and $y$ are digits such that $17! = 355687428096000$,then $x+y$ equals

The number of four-letter words that can be formed using the letters of the word $BARRACK$ is

In how many ways can $3n$ distinct objects be distributed equally among $3$ persons?

$A$ man has $7$ relatives,$4$ of them are ladies and $3$ are gents; his wife has $7$ other relatives,$3$ of them are ladies and $4$ are gents. The number of ways they can invite $3$ ladies and $3$ gents to a party such that there are $3$ of the man's relatives and $3$ of the wife's relatives,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 . . . . . . .