Expert: Paul Klarreich - 9/24/2008

Question
Suppose you have members of six families which each consist of a mom, a dad, and a kid.

How many ways can the families sit in a row of 18 chairs such that the members of each family are sitting together.  

I realize that there are six possible arrangements of the the three family members which must sit together.  What is confusing me is regarding how to deal with the situation of the chairs.

Thank you so much for you help!

Answer
Questioner:   Jolene
Category:  Advanced Math
Private:  No
 
Subject:  Permutation Problem
Question:  Suppose you have members of six families which each consist of a mom, a dad, and a kid.

How many ways can the families sit in a row of 18 chairs such that the members of each family are sitting together?

I realize that there are six possible arrangements of the the three family members which must sit together.  What is confusing me is regarding how to deal with the situation of the chairs.

Thank you so much for you help!
....................................................
Hi, Jolene,

Start this way:

How many ways can the families sit in a row of SIX SOFAS? (naturally, one sofa per family.)

Obviously that is 6!   (Factorial of six, that is.  This stupid web site mangles special characters, so I am not sure it comes through properly.)


Next replace the first sofa with three chairs, permuting the first family. -->  3! ways, of course.
Same for each of the others.  Now apply the multiplication principle.

Total = 6! (3!)^6 = some large number.