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A circular garden divided into 10 equal sectors needs to be planted with flower plants that yield flowers of 3 different colors, in such a way that no two adjacent sectors will have flowers of the same color. The number of ways in which this can be done is
Correct Answer: 1026
Colouring the sectors of a circle so that neighbours differ is the same as properly colouring a cycle with 10 vertices. The number of ways with $$k$$ colours is $$(k-1)^n + (-1)^n (k-1)$$. With $$n = 10$$ and $$k = 3$$ this is $$2^{10} + 2 = 1026$$.
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