Problem 13 - Thinking Mathematically

Question

A person can order a new car with a choice of six possible colors, with or without air conditioning, with or without automatic transmission, with or without power windows, and with or without a CD player. In how many different ways can a new car be ordered with regard to these options?

Step-by-Step Solution

Answer

The total number of different ways a new car can be ordered with regard to these options is \(6 \times 2 \times 2 \times 2 \times 2 = 96\) different ways.

Step 1: Count the choices of car color

For the color of the car, there are 6 options. This means there are 6 possible choices for the color of the car.

Step 2: Count the choices for having or not having air conditioning

For air conditioning, there are 2 options: having air conditioning or not having air conditioning. So, there are 2 possible choices for air conditioning.

Step 3: Count the choices for automatic or manual transmission

For automatic transmission, there are 2 options: the car can have an automatic transmission or not. Thus, there are 2 possible choices for the type of transmission.

Step 4: Count the choices for power or non-power windows

For power windows, there are again 2 options: the car can have power windows or not. Therefore, there are 2 possible choices for the type of windows.

Step 5: Count the choices for having or not having a CD player

For the CD player, like for the previous features, there are 2 choices: either a CD player is included or it is not. So, there are 2 possible choices for the CD player.

Step 6: Calculate total number of possible orders

To get the total number of possible car orders, multiply the number of choices in each category. Therefore, the total number of different orders is \(6 (colors) \times 2 (air conditioning choices) \times 2 (transmission choices) \times 2 (window choices) \times 2 (CD player choices) \).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many practical applications ranging from computer science to statistical physics. The problem at hand illustrates a fundamental principle called the counting principle, which states that if there are 'n' ways to do something and 'm' ways to do another thing after that, there are 'n × m' total ways of performing both actions.

For example, when ordering a car, there are multiple independent choices to be made, such as the color or the accessories like air conditioning. According to combinatorics, each independent choice multiplies the total number of possibilities. If you imagine each option as a branch on a tree, each consecutive choice creates a new set of branches, resulting in a tree with many possible outcomes or combinations.

Permutations and Combinations

When people refer to permutations and combinations, they are typically talking about ways to arrange or select items where order matters (permutations) or does not matter (combinations). Even though this particular problem does not directly ask for permutations or combinations, understanding them can provide insight into more complex problems.

In this scenario, having or not having a feature such as a CD player is a simple example of combinations. You are not concerned with the order the features are selected, only whether or not they are included. Calculating the total configurations of features for a new car is similar to solving a problem based on combinations where order doesn't matter but selection does. Highlighting the role of these concepts in everyday decisions can demystify the subject and make it more relatable to students.

Mathematical Reasoning

Mathematical reasoning involves the ability to think logically about quantitative and spatial information to draw conclusions. It's what allows us to take a step back and look at a problem, break it down into parts, and solve it step by step -- which is exactly what's done in the car ordering example. Every time we count the different options and multiply them together, we're using mathematical reasoning to understand how choices compound.

In the context of ordering a new car, we reason that the choice of one feature (like air conditioning) does not affect the other (like color), allowing us to multiply the number of options together to get the total. Mathematical reasoning is the engine behind recognizing patterns, making connections, and solving complex problems effectively. It allows students to approach mathematics not just as a series of numbers or equations, but as a logical, interconnected system that can be applied to real-world situations.