GRE 5- There are six different models that are to appear in a fashion show.

There are six different models that are to appear in a fashion show. Two are from Europe, two are from South America, and two are from North America. If all the models from the same continent are to stand next to each other, how many ways can the fashion show organizer arrange the models?
A) 48
B) 64
C) 24
D) 8
E) 72

1. In this problem, order is important. Jane standing to the left of Mary is a different arrangement than Mary standing to the left of Jane. Because order is important, the permutations equation is used.
2. The formula for a permutation is:
   _nP_r= n!/(n-r)!
   where n is the total number of selections available and r is the number of items to be selected.
3. For each set of two models from each continent, there are ₂P₂ways to arrange them. From the permutation formula with n equal to 2 and r equal to 2 this results in:
   ₂P₂= 2! = 2
4. Since there three groups of two there are:
   ₂P₂*₂P₂*₂P₂ or 2*2*2 or 8 ways to arrange each group within each group.
5. Since there are three groups of models that can be placed in three different positions there are:
   ₃P₃ways to arrange the three groups.
6. The value of₃P₃is (3*2*1)/(3-3)! = 6/0! = 6, or 6 ways to arrange the different groups. Note that 0! is defined to be equal to one.
7. Since there are 6 ways to arrange the groups and 8 ways to arrange the models within their own groups there are:
   8X6 or 48 different ways to arrange the models. So the correct answer is A.