Mock AIME 1 2007-2008 Problems/Problem 3

Problem

A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?

Solution

We apply the complement principle: we find the total number of cases in which the 2 green places are adjacent, and subtract from the total number of cases.

There are ways to arrange the plates in a linear fashion. However, since the plates are arranged in a circle, there are ways to rotate the plates, and so there are ways to arrange the plates in a circular fashion (consider, for example, fixing the orange plate at the top of the table).

If the two green plates are adjacent, we may think of them as a single entity, so that there are now objects to be placed around the table in a circular fashion. Using the same argument, there are ways to arrange the objects in a linear fashion, and ways in a circular fashion.

Thus, the answer is .

See also

Mock AIME 1 2007-2008 (Problems, Source)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15