Problem 21 - College Algebra

Question

Use the Fundamental Counting Principle to solve Exercises \(21-32\). 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 you can order the car.

Step 1: Identify the variables

The given variables are: \n Number of colors (C) = 9 \n Number of styles (S) = 3

Step 2: Apply the Fundamental Counting Principle

When making a selection of both color and style, the number of ways that both selections can be made at the same time according to the Fundamental Counting Principle is the product of the number of ways each can be made. In this case, that is the product of the number of colors and the number of styles or C * S.

Step 3: Compute the total

Substitute the values of C and N into the formula from Step 2 to solve for the total number of ways: Total = C * S = 9 * 3

Key Concepts

Permutations

Permutations form the foundation of the Fundamental Counting Principle and are essential in solving many problems in combinatorics, the branch of mathematics that deals with counting, arrangement, and combination in sets.

A permutation is an arrangement of objects in a specific order. The number of permutations is determined by how many ways you can order a set of items. When the order matters in the arrangement, that's when the concept of permutations kicks in.

For example, if you're choosing a password with four distinct digits, the arrangement 1234 is different from 4321 – both are unique permutations. The Fundamental Counting Principle facilitates the calculation of permutations by allowing us to simply multiply the number of choices for each category or step. If a car model comes in nine colors and three styles, we multiply 9 (the number of colors) by 3 (the number of styles) to find the number of distinct permutations for ordering the car.

Combinatorics

Combinatorics is the field of mathematics that specializes in counting and analyzing combinations and arrangements of objects. Within combinatorics, we use the Fundamental Counting Principle as one of the most basic and powerful tools.

The principle states that if you have a series of choices or events, and each one has a finite number of ways it can occur, the total number of ways for the entire series to occur is the product of the number of ways each individual choice or event can occur.

This principle is invaluable when dealing with larger, more complex problems because it simplifies the process significantly by breaking it down into smaller, more manageable parts. It's often the starting point before getting into more nuanced ideas like permutations and combinations, where additional rules regarding order and repetition come into play.

Probability

Probability is a measure of how likely an event is to occur out of the number of possible outcomes. It plays a central role in combinatorics and permutations, as it often involves calculating the likelihood of different arrangements or selections.

The theory of probability ties in with the Fundamental Counting Principle which helps determine the number of favorable outcomes, thus facilitating the calculation of probability as a ratio of favorable to total possible outcomes. For instance, if you want to know the probability of selecting a red hatchback from the car models available which come in nine colors and three styles, you would first count the permutation of a red hatchback (which is 1, since there's only one way to choose a red hatchback) and then divide it by the total number of car permutations (which we've calculated using the Fundamental Counting Principle to be 27).

It's important to note that probability values range from 0 (impossible event) to 1 (certain event). Expressing probability as a percentage, fraction or decimal is common, although context determines the most appropriate form.