Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

A customer can choose one of four amplifiers, one of ten stereo receivers, and one of five speaker models for an entertainment system. Determine the number of possible system configurations.

Step-by-Step Solution

Answer

The total possible system configurations are 200.

Step 1 - Calculate Amplifier Choices

There are 4 amplifier choices. So, the number of amplifier choices is 4.

Step 2 - Calculate Stereo Receiver Choices

There are 10 stereo receiver choices. So, the total number of stereo receiver choices is 10.

Step 3 - Calculate Speaker Model Choices

There are 5 speaker model choices. So, the total number of speaker model choices is 5.

Step 4 - Calculate Total Configurations

Now, to find the total possible system configurations, multiply the number of choices for each of the three components together. So, the total configurations = amplifier choices x stereo receiver Choices x speaker model choices = 4 x 10 x 5

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of elements within a set. It often involves finding the total number of possible outcomes without having to list them all explicitly. This area of math is incredibly useful in various fields such as computer science, probability, and optimization.

In the context of our exercise, combinatorics is applied to find the number of possible entertainment system configurations based on a selection of amplifiers, stereo receivers, and speaker models. It uses fundamental counting principles that allow us to calculate the number of combinations efficiently.

Permutations and Combinations

Permutations and combinations are two core concepts within combinatorics that deal with the arrangement of items. A permutation is an ordered arrangement of items, where the order matters. On the other hand, a combination is an arrangement where order is irrelevant.

For our problem, it is essential to recognize that the order in which the customer chooses each component (amplifier, receiver, and speaker) does not change the configuration. Hence, we are dealing with combinations here. Each component is chosen independently, but the overall system configuration does not rely on the order of selection. This allows us to simply multiply the number of choices for each component to find the total number of unique configurations.

Problem-solving Strategies

Effective problem-solving strategies are crucial in tackling complex combinatorial problems. One common strategy is to break down the problem into smaller, more manageable parts, as seen in the step-by-step solution.

Firstly, determine the number of choices for each category separately. Then, instead of combining them all at once, calculate the possible options for each category (amplifiers, receivers, speakers), and then use a counting principle to bring it all together. Such strategies help simplify the process and ensure that all potential combinations are accounted for without confusion or redundancy.

Counting Principles

Counting principles form the foundation upon which combinatorics is built. The most basic counting principle is the Fundamental Principle of Counting, which states that if one event can happen in 'm' ways and a second independent event can happen in 'n' ways, then together they can happen in 'm x n' ways.

This principle is directly applied to our exercise. Since the choice of an amplifier, stereo receiver, and speaker model are independent events, the total configurations can be found by multiplying the number of choices for each: 4 (amplifiers) x 10 (receivers) x 5 (speakers), resulting in a total of 200 possible system configurations. Understanding and applying this principle correctly is a key skill in solving combinatorial problems.