Problem 2 - Thinking Mathematically

Question

The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or station wagon). In how many ways can you order the car?

Step-by-Step Solution

Answer

There are 27 different ways to order the car.

Step 1: Identify the number of ways to choose a color

The problem states that the car model is available in nine different colors. Thus, there are nine possible ways to choose a color.

Step 2: Identify the number of ways to choose a style

The problem states that the car model is available in three different styles (hatchback, sedan, or station wagon). Thus, there are three possible ways to choose a style.

Step 3: Use the counting principle to find the total number of ways

According to the counting principle, we multiply the number of ways to choose a color by the number of ways to choose a style. So, 9 (ways to choose a color) multiplied by 3 (ways to choose a style) give us 27. Therefore, there are 27 different ways to order the car.

Key Concepts

Permutations

When we talk about permutations, we're discussing the different ways in which a set of things can be arranged or ordered. However, in the given car ordering scenario, it's important to note that permutations specifically deal with order. For example, if you are selecting the order of colors or styles, permutations would be the tool you use if the sequence does matter. In contrast, our car problem is more about combinations, because the order of choice—color or style—doesn't impact the outcome.It's crucial to differentiate permutations from simple combinations. Permutations would ask how many different ways you can list colors and styles, but that's not necessary here since any chosen combination is unique on its own. If we had a permutation-based problem, we would calculate using factorials, as shown:

• With permutations, the formula involves:
  • For example: selecting 2 colors out of 3 real options—where order matters—would be computed using this formula.
  • The calculation would look like: \( 3! \), which translates into \( 3 \times 2 \times 1 = 6 \) different permutations.

Understanding permutations encourages you to look at problems where his order plays a key role.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of sets. It provides us with strategies to organize and count possibilities in structured ways. In the car selection example, combinatorics helps us calculate all possible outcomes without getting overwhelmed. This problem is an illustration of the fundamental counting principle, an integral part of combinatorics. This principle assists us in determining how many ways multiple choices can occur together. For instance, the car can be any combination of color and style.

• 9 color choices × 3 style choices = 27 possible combinations

Thanks to combinatorics, planning and determining these possibilities become manageable. Instead of listing everything manually, we effectively apply structured mathematical thinking, like the counting principle, to answer such questions effortlessly.

Problem Solving

Problem solving is about understanding the problem at hand and systematically working through to find a solution. In the context of the text problem, we begin by breaking it down into smaller parts to simplify counting, using clear approaches. First, identify what needs choosing—the color and the style in this case. Then apply a well-known strategy, the counting principle, which allows us to easily multiply choices together. Here’s how problem-solving played a key role:

• Recognize: Identify that each choice (color or style) is independent and needs to be determined.
• Explore: Apply methods like the counting principle to multiply choices.
• Implement: Execute the plan. For instance, multiplying 9 possible colors by 3 available styles gives 27 distinct options for ordering a car.

Efficient problem solving often involves breaking problems into more manageable steps, just as seen here. Approaches used here can be applied to other areas where multiple independent options need counting. Simplifying complexity into steps can transform seemingly tough problems into straightforward solutions.