# Math Help - Permutations of people

1. ## Permutations of people

Hi all, I've got this question here... don't even know how to begin solving it.

There are 28 people from 7 countries; 4 people from each country. I'm required to put them in a line, so that each person will have at least one neighbour from the same country. How many such orderings exist?

Thanks in advance.

2. Originally Posted by Aldarion
There are 28 people from 7 countries; 4 people from each country. I'm required to put them in a line, so that each person will have at least one neighbor from the same country. How many such orderings exist?
Suppose that {a,b,c,d} is the set of people from country Q.
Then there are three parings of the four people: (ab)(cd); (ac)(bd); (ad)(bc).
Thus there are $3^7$ ways to choose the paring of all 28 people.

For any choice of parings, we have $(14!)$ ways to arrange these pairs into a line.

In each of those lines there are $2^{14}$ ways to arrange the couples.