Find the number of different $8$-letter arrangements that can be made from the letters of the word $DAUGHTER$ such that all vowels do not occur together.

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

$n$-digit numbers are formed using only three digits $2, 5,$ and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed is:

There are $9$ chairs in a room on which $6$ persons are to be seated. Out of these,one person is a guest who must occupy one specific chair. In how many ways can they sit?

If all the letters of the word $REPEAT$ are permuted in all possible ways and if the six-letter permutations thus formed are arranged in the dictionary order,then the rank of the word $REPEAT$ is:

All the letters of the word $LETTER$ are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order. Then the rank of the word $TETLER$ is