In how many ways can $5$ prizes be distributed among four students when every student can take one or more prizes?

Using the digits $0, 2, 4, 6, 8$ not more than once in any number,the number of $5$-digit numbers that can be formed is:

The letters of the word $MODESTY$ are written in all possible orders and these words are written out as in a dictionary. Then,the rank of the word $MODESTY$ is:

$_{n-1}P_r + r \cdot (_{n-1}P_{r-1}) = \dots$

How many different nine-digit numbers can be formed from the digits of the number $223355888$ by rearranging the digits such that the odd digits occupy even places?

$5$ 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 $\alpha: \beta=$