$5$ 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 $\alpha: \beta=$

The number of different nine-digit numbers that can be formed by rearranging the digits of the number $223355888$ such that the odd digits occupy even positions is:

Using the letters of the word $TRICK$,a five-letter word with distinct letters is formed such that $C$ is in the middle. In how many ways is this possible?

It is required to seat $5$ men and $4$ women in a row so that the men occupy odd places. Then the number of arrangements that are possible is

The number of ways in which the following prizes can be given to a class of $20$ boys,first and second Mathematics,first and second Physics,first Chemistry,and first English is:

Each of the $10$ letters $A, H, I, M, O, T, U, V, W$ and $X$ appears the same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many $3$-letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?