91Ó°ÊÓ

In mathematics, a factorial is a function that multiplies a number by all of the positive integers below it. Represented by the symbol \(!n\), the factorial of a non-negative integer \(n\) is defined as \(!n = n \times (n-1) \times (n-2) \times ... \times 1\). For example, the factorial of 4 is \(!4 = 4 \times 3 \times 2 \times 1 = 24\).
Factorials grow rapidly with larger numbers. For 11 trees in our problem, we compute \(!11 = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39916800\).
This massive number elucidates how quickly factorials can increase.

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of elements in sets. It's crucial in solving problems related to permutations and combinations.
In our tree planting problem, combinatorics helps us determine how many ways we can arrange the trees along a fairway. By calculating permutations, we identify every possible way to line up 11 trees if they were unique. However, because some trees are identical, we must adjust our calculations for these repetitions.
Using combinatorics, we divided by the factorial of each group of identical trees to avoid over-counting arrangements. This step ensures we only count unique permutations.

When dealing with permutations, identical objects require special consideration. If an arrangement contains multiple identical items, simply calculating the factorial of the total number won't yield the correct answer.
In our problem, we have three groups of identical trees: 4 linden trees, 5 white birch trees, and 2 bald cypress trees. To appropriately account for these repetitions, we adjust our formula:
\[ \text{Number of unique arrangements} = \frac{11!}{4! \times 5! \times 2!} \]
This division eliminates the redundant permutations where identical trees switch places, giving us only the distinct arrangements.

To determine the total arrangements of trees along the fairway, we start by considering how we'd arrange all trees if each were unique. This calculation uses the factorial function, where for the 11 trees, it’s \(!11\).
However, given the presence of identical trees, these identical arrangements get over-counted. We correct this by dividing by the factorial of the number of each type of tree:
\[ \frac{11!}{4! \times 5! \times 2!} \]
Inserting our specific values, the computation is \[ \frac{39916800}{24 \times 120 \times 2} = 6930 \. \]
This adjustment ensures we only count the unique ways to line up the trees, resulting in 6930 distinct arrangements. Thus, the landscaper has 6930 options for planting the trees in a row.