There are six chairs at a round table. A seating arrangement will be considered the same if everyone at the table has the same neighbor to the left to the right.
a. how many different ways can we seat six people, assuming that one person sits in each chair?
b. how many different ways can we seat four people, assuming that two chairs are left empty?
c. how many different ways can we seat three couples if every person is seated next to their date?
a. let's pick one person, and fill the seats clockwise.
There will be 5 possibilities for his left neighbor. then 4 possibilities for the next seat, then 3 possibilities for the next seat, then 2 possibilities then 1.
The number of different ways is 5!=120
b. Let's pick one person then seat the 3 other people and finally add 2 empty chairs between the people.
There will be 3!=6 different ways to seat the 4 people in a chair.
Now for each configuration of 4 people, we may add 2 seats between the same 2 people or between 2 different couple of people.
If the two empty seats are next to each other, there are 4 possibilities. If the empty seats are between 2 different pairs, there are 4*3/2 possibilities. Therefore given each configuration of 4 people there are 4+6=10 ways of displaying the empty chairs.
Therefore there are 6*10=60 ways of displaying 4 people in a circle of 6 chairs.
c)Let's pick one couple. One couple will be on the left and the other on the right. Which makes 2 possibilities.
Now each couple can decide to have the male either on the right or on the left. It multiplies the number of possibilities by 2^3=8
Therefore there are 2*8-16 ways to place 3 couples around the table.