How many distinct words can be formed using the letters of the word $ARRANGE$?

$5$ men and $4$ women are seated in a row. If the number of arrangements in which one particular man and one particular woman are together is $\alpha$ and the number of arrangements in which those two are not together is $\beta$,then $\alpha: \beta=$

Three children,each accompanied by a guardian,seek admission in a school. The principal wants to interview all the $6$ persons one after the other subject to the condition that no child is interviewed before its guardian. In how many ways can this be done?

In how many ways can $8$ people stand in a line such that there are always exactly two people between two specific people $A$ and $B$?

How many numbers lying between $999$ and $10000$ can be formed with the help of the digits $0, 2, 3, 6, 7, 8$ when the digits are not to be repeated?

How many numbers less than $1000$ can be formed using the digits $0, 1, 2, 4,$ and $5$ if repetition of digits is not allowed?