Mock AIME 1 2007-2008 Problems/Problem 3

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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