Problem

A golf-course architect has six linden trees, five white birch trees, and three bald cypress trees to plant in a row along a fairway. In how many ways can the landscaper plant the trees in a row, assuming that the trees are evenly spaced?
The trees-can be planted in different ways.

Answer

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Answer

Final Answer: The landscaper can plant the trees in \(\boxed{168168}\) different ways.

Steps

Step 1 :The problem is a permutation problem with repetition. The total number of trees is 6 (linden) + 5 (white birch) + 3 (bald cypress) = 14 trees. Since the trees of the same type are indistinguishable, we need to divide the total permutations by the repetitions of each type of tree.

Step 2 :First, calculate the total permutations of 14 trees, which is \(14!\) or 87178291200.

Step 3 :Next, calculate the permutations of each type of tree. The permutations of 6 linden trees is \(6!\) or 720, the permutations of 5 white birch trees is \(5!\) or 120, and the permutations of 3 bald cypress trees is \(3!\) or 6.

Step 4 :Finally, divide the total permutations by the permutations of each type of tree. The number of ways the landscaper can plant the trees is \(\frac{87178291200}{720 \times 120 \times 6}\) or 168168.

Step 5 :Final Answer: The landscaper can plant the trees in \(\boxed{168168}\) different ways.

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