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...

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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.**