Problem 30 - Discrete Mathematics with Applications

Question

A botanist would like to plant three coleus, four zinnias, and five dahlias in a row in her front garden. How many ways can she plant them if: They can be planted in any order.

Step-by-Step Solution

Answer

The botanist can plant the 12 plants (3 coleus, 4 zinnias, and 5 dahlias) in a total of \(\frac{12!}{3!4!5!}\) different ways, since there are repetitions of each type of plant.

Step 1: Understand the Problem

We are given:
A botanist would like to plant three coleus, four zinnias, and five dahlias in
a row in her front garden. How many ways can she plant them if:
They can be planted in any order.

Step 2: Apply Relevant Concepts

We apply mathematical definitions, properties, and theorems.

Step 3: Solution

The botanist can plant the 12 plants (3 coleus, 4 zinnias, and 5 dahlias) in a total of \(\frac{12!}{3!4!5!}\) different ways, since there are repetitions of each type of plant.

Key Concepts

Factorials

A factorial, denoted by an exclamation mark such as in \( n! \), is a fundamental concept in mathematics. It represents the product of all positive integers up to a given number \( n \). For example, the factorial of 5 is calculated as follows:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials are essential in counting problems and appear frequently in permutations and combinations. This is because factorials are used to determine how many ways things can be arranged or grouped.
In our exercise, we calculate the factorial of 12, which includes all possible arrangements of 12 plants: 3 coleus, 4 zinnias, and 5 dahlias. The factorial \( 12! \) looks like this:

• \( 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)

The result of this calculation provides the total number of ways to arrange the 12 diverse plants in the garden. Understanding how to compute factorials is crucial, especially in problems involving arrangements and permutations.

Combinatorics

Combinatorics is a branch of mathematics focused on counting and arrangement possibilities. It examines how items can be selected and arranged subject to specific constraints.
This field is vast and encompasses different topics like permutations, combinations, and more complex arrangements with repetition or restrictions.
In the given problem, the botanist's task involves using ideas from combinatorics to understand how multiple types of plants can be arranged in various sequences.

• Permutations are a combinatorial method to count arrangements where order matters.
• Combinations factor in situations where order doesn’t matter.

For our exercise, we're dealing with permutations because the order in which the plants are arranged directly affects the outcome. The factorial approach is a tool within combinatorics that simplifies the process of counting these permutations efficiently.
By using combinatorics, we can determine the number of distinct planting sequences, ensuring not a single possible arrangement is missed.

Arrangement

When we talk about arrangements in mathematics, especially in the context of permutations, we refer to the specific placement order of items. Arrangement focuses on how to systematically organize or lay out objects, respecting certain criteria or preferences.
In the example with the botanist planting a garden, the "arrangement" is critical to understand how the plants are lined up.
For this specific question:

• The arrangement involves mixing different plant types: coleus, zinnias, and dahlias.
• Each distinct sequence of the plants counts as a separate arrangement.
• Since there are no restrictions given, any order of plants is valid.

The significance of solving arrangement problems lies in its application in real-world settings, such as designing layouts, planning sequences, or organizing schedules.
Recognizing the importance of arrangement helps in visualizing and planning efficient solutions to combinatorial problems, like organizing the plants in the garden as needed by the botanist without any limitations.