In how many ways can the letters of the word $PERMUTATIONS$ be arranged if there are always $4$ letters between $P$ and $S$?

The number of $4$-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word $INCONSEQUENTIAL$, without repeating any letter, is:

$A$ number lock consists of three rings. If the $1^{st}$ ring is marked with digits $0$ to $9$,the $2^{nd}$ ring is marked with prime numbers greater than $2$ but less than $30$,and the $3^{rd}$ ring is marked with all vowels,find the total number of unsuccessful attempts.

In how many ways can $5$ girls and $3$ boys be seated in a row so that no two boys are together?

If the different permutations of all the letters of the word $EXAMINATION$ are listed as in a dictionary,how many words are there in this list before the first word starting with $E$?

$a, b, c$ are three particular speakers among the $10$ speakers of a meeting. The number of ways of arranging all the $10$ speakers on the dais in a row so that all the three speakers $a, b, c$ do not sit together is