Expert: Steve Holleran - 4/2/2008

Question
Three couples, six individuals, sit down randomly around a round table.  What is the probability that at least one couple will be seated next to each other?

Answer
Hi Ken,

I am definitely not very good at these probability problems, but here's the only ideas I have, after working on this for three days:

P(at least 1 couple seated together) =

                  1 - P(no couples seated together).

If you call the couples a1, a2  ;   b1,b2  ;  c1,c2

and the x's are the seats:

                            x


                    x                x


                    x                 x


                             x

If you start at the top, there are 6 choices for the first seat.  Then you only have 4 choices for the next seat to the right (going clockwise around the table), you have 3 choices for the third seat, then 2 for the fourth, and 1 each for the other two.  This gives
144 ways of seating.

There are (6-1)! = 120 ways to seat 6 people around a table in one direction, so that makes 240 for both directions.

Then we have :

       1 - P(none together) = 1 - 144/240 = 96/240 = .4 = 40%

I haven't done this stuff in years, and I have forgotten a lot (obviously)

Hope some of it helps out, anyway.
Steve