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?

1 Answer
Mar 21, 2016

Answer:

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.
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