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?

Mar 21, 2016

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:
$\textcolor{w h i t e}{\text{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.