a golf - course architect has four linden trees, four white birch trees, and three bald cypress trees to…

a golf - course architect has four linden trees, four 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.

a golf - course architect has four linden trees, four 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

Explanation:

Step1: Calculate the total number of trees

The total number of trees is (4 + 4+3=11) trees.

Step2: Use the formula for permutations of multi - sets

The formula for permutations of a multi - set with (n) objects, where there are (n_1) of one kind, (n_2) of another kind,(\cdots), (n_k) of the (k)th kind is (\frac{n!}{n_1!n_2!\cdots n_k!}). Here (n = 11), (n_1=4) (linden trees), (n_2 = 4) (white birch trees) and (n_3=3) (bald cypress trees). [ \begin{align*} \frac{11!}{4!4!3!}&=\frac{11\times10\times9\times8\times7\times6\times5\times4!}{4!\times4\times3\times2\times1\times3\times2\times1}\ &=\frac{11\times10\times9\times8\times7\times6\times5}{4\times3\times2\times1\times3\times2\times1}\ &=\frac{1663200}{288}\ & = 5775 \end{align*} ]

Answer:

(5775)