How many ways can a gardener plant five different species of shrubs in a circle? What...

How many ways can a gardener plant five different species of shrubs in a circle? What is the answer if two of the shrubs are the same? What is the answer if all the shrubs are identical?

Solutions

Expert Solution

Let be the different species of shrubs to be planted in five different places on a circle.

has five choices to get planted. has four choices, has three, has two and has only one choice to get planted. So, total number of structures (with repetition) we get on a circle is . But here, every structure is repeated times. So the total number of different structures i.e. number of ways of planting these shrubs on a circle is .

Now, suppose two shrubs, say and , where , are of the same species. So any two structures with and interchanging their places will be the same. Therefore Number of ways in this case is .

When all the shrubs are the same, naturally there is only one way to plant them on a circle.

Colby Messinger answered 1 year ago

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