Question

A​ golf-course architect has four linden​ trees, five white birch​ trees, and twotwo 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?

Answer 1

Given, there are,

4 linden trees,

5 brich trees,

And 2 bald cypress trees,

So, total number of trees to be planted is (4+5+2)=11 trees,

Now these 11 trees can be arranged( or planted) in a row (evenly spaced) in 11! ways.

But among these 11! possible arrangements, the 4 linden trees can arrange themselves in 4! ways, but since all the linden trees are identical, the ordering among the linden trees doesn't matter, similarly 5 white brich trees can be arranged among themselves in 5! ways, which also doesn't matter and lastly the 2 bald cypress trees can arrange among themselves in 2! ways, which also doesn't matter.

So, in total the trees of each type can be arranged amon themselves in 4!×5!×2! ways, which are to be ignored among the all possible 11! ways of arrangements,

So, the all possible ways the 11 trees (consisting of 4 linden, 5 white brich and 2 bald cypress) can be planted(or arranged) in a row (spaced evenly) in

So, all the trees xan be planted in a row spaced evenly in 6930 ways. Hence the answer.