If ${}^n P_4 = 1680$,then $n =$

The number of permutations by taking all letters and keeping the vowels of the word $COMBINE$ in the odd places 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 $\alpha: \beta=$

There are $6$ periods in each working day of a school. The number of ways one can arrange $5$ subjects such that each is allowed at least one period is:

How many ways can you arrange all the characters in "$KCET$ $2025$" such that the arrangement starts with $K$ and ends with $5$?

The number of ways in which $6$ distinct rings can be worn on the $4$ fingers of one hand is