Find the number of arrangements of the letters of the word $INDEPENDENCE$. In how many of these arrangements do the vowels never occur together?

The numbers $1, 2, 3,$ and $4$ are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. $A$ person draws two slips from the box,one after the other,without replacement. Describe the sample space for the experiment.

There are $10$ students in a classroom,including $3$ girls $A, B,$ and $C$. In how many ways can they be arranged in a row such that no two of the three girls are together?

How many $3$-digit even numbers can be formed using the digits $1, 2, 3, 4, 6, 7$,if no digit is repeated?

If the letters of the word $SACHIN$ are arranged in all possible ways and these words are written out as in a dictionary,then the word $SACHIN$ appears at serial number:

If $^n{P_5} = 9 \times {^{n - 1}}{P_4}$,then the value of $n$ is