Question

How many ways can you seat `5` gentlemen and `3` ladies around a circular table with the requirement that every gentleman is seated next to at least one lady?

Answer 1

Depending upon the interpretation,
either
`color(white)("XXX")5! * 3! =720`
or
`color(white)("XXX")8 * 5! * 3! = 5760`

Explanation:

Starting at some seating position the pattern must be:
`color(white)("XXX")`2 gentlemen - 1 lady - 2 gentlemen - 1 lady - 1 gentleman - 1 lady

If different starting locations for this pattern are considered the same seating pattern:
There are `5!` ways of distributing the gentlemen among their seats
and for each of these, there are `3!` ways of distributing the ladies among their seats
for a total of `5! * 3!` different seating patterns.

If the starting location for this pattern is considered to give a different way of seating then
there will be `8` times as many seating patterns.