$A$ number lock has $3$ rings and each ring has $8$ digits. The total number of different ways in which the $3$ rings can be rotated is:

The number of $4$ letter words that can be formed with the letters in the word $EQUATION$ with at least one letter repeated is

If $^mP_r - ^{m-1}P_r = a \cdot ^{m-1}P_s$,then $a - s = \dots$

The number of ways of arranging the letters of the word $LINEAR$ so that the letters $N$ and $R$ do not come together and $E$ and $A$ come together 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 $\alpha: \beta=$

In how many ways can $mn$ letters be posted in $n$ letter-boxes?