There are $5$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back,is

$5$ 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 $\alpha: \beta=$

How many distinct words can be formed using the letters of the word $EQUATION$ such that the word starts and ends with a consonant?

If the letters of the word $MOTHER$ are arranged in all possible ways and these words are listed in dictionary order,what is the rank of the word $MOTHER$?

$A$ lock is made with $3$ rings,each marked with $10$ different letters. How many unsuccessful attempts can be made to open the lock?

How many $4$-digit numbers can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ such that each digit is used only once?