Question

The contractor wants to plant six oak trees, nine maple trees, and five poplar trees along the subdivision street. The trees are to be spaced evenly. In how many distinguishable ways can they be planted?

Answer 1

To find the number of distinguishable ways to plant the trees, calculate the factorial of the total number of trees and divide by the product of the factorials for each type of tree.

Step-by-step explanation:

The question asks about the number of distinguishable ways to plant six oak trees, nine maple trees, and five poplar trees with even spacing along a subdivision street. To solve this, we use the concept of permutations of multiset. To find the number of distinguishable arrangements, we divide the factorial of the total number of trees by the product of the factorials of each type of tree.

Total number of trees = 6 (oak) + 9 (maple) + 5 (poplar) = 20 trees.

Number of distinguishable ways to plant them = \(\frac{20!}{6! \times 9! \times 5!}\).

This calculation considers that trees of the same species are indistinguishable from one another. The factorial notation (!) represents the product of all positive integers up to that number.