Problem 70 - Basic Statistics for Business and Economics with Student

Question

Reynolds Construction Company has agreed not to erect all "look-alike" homes in a new subdivision. Five exterior designs are offered to potential home buyers. The builder has standardized three interior plans that can be incorporated in any of the five exteriors. How many different ways can the exterior and interior plans be offered to potential home buyers?

Step-by-Step Solution

Answer

15 unique combinations are possible.

Step 1: Identify the Choices

First, recognize that the exercise involves two independent choices for the home design: exterior designs and interior plans.

Step 2: Determine Choices for Each Component

There are 5 exterior designs available and 3 different interior plans that can be used with each of those exteriors.

Step 3: Compute Total Combinations

To find the total number of combinations of exteriors and interiors, multiply the number of exterior designs by the number of interior plans: \[ 5 \times 3 = 15 \]

Step 4: Conclusion: Interpret the Result

This result means there are 15 unique ways to combine the available exteriors and interiors to offer to potential home buyers.

Key Concepts

Independent Choices in Combinatorics

In combinatorics, the idea of independent choices is crucial to solving complex problems. Essentially, independent choices refer to decisions that do not affect one another.
For example, when Reynolds Construction Company offers five exterior designs and three interior plans, each choice acts independently of the others.
This means choosing an exterior doesn't limit or influence the choice of interiors.

• Exterior designs can vary independently of what's inside.
• Interiors can be tailored regardless of the external appearance.

Understanding this independence allows us to confidently calculate the number of possible combinations without needing to worry about overlapping or interfering criteria.

Multiplication Principle Explained

The multiplication principle is a fundamental rule in combinatorics. It helps us determine the total number of outcomes for a sequence of independent choices. When faced with a series of decisions, you multiply the number of options available for each decision to find the total combinations.
In the exercise, this is how it works:

• Reynolds has 5 choices for exteriors.
• Each exterior can be combined with any of the 3 interior plans.

Simply multiply the number of choices for each independent decision: \(5 \times 3 = 15\).

Whenever tasks are performed independently in sequence, use the multiplication principle for simplicity and accuracy.

It ensures that every possibility is accounted for, providing an exhaustive view of the options.

Understanding Unique Combinations

Unique combinations are the sets of outcomes that arise from different configurations without repetition. In the subdivision example, every pairing of an exterior with an interior results in a unique combination.

• Each house can have a distinct appearance based on these pairings.
• No two combinations are the same, making each offering special.

This uniqueness is often required in scenarios to provide variety or customization, such as building homes.
Recognizing unique combinations helps in offering diverse and non-repetitive choices to customers, ensuring a wide array of options that cater to different preferences and needs.