Problem 9 - Algebra 2

Question

Automobiles You should rotate tires on a car at regular intervals. a. In how many ways can four tires be arranged on a car? b. If the spare tire is included, how many arrangements are possible?

Step-by-Step Solution

Answer

There are 24 different ways to arrange four tires on a car without the spare tire and 120 ways when including the spare tire.

Step 1: Understanding the Problem

We are asked to determine the number of different ways in which four tires can be arranged on a car. This is a permutation problem since the order in which the tires are placed matters.

Step 2: Calculating Arrangements without the Spare Tire

To calculate the number of different arrangements of four tires, we use the permutation formula without repetitions, which is simply the factorial of the number of tires, or 4!.

Step 3: Executing the Calculation for Four Tires

Compute 4! (4 factorial), which is 4 x 3 x 2 x 1.

Step 4: Understanding the Second Part of the Problem

Now, we include the spare tire, increasing the number of tires to arrange to five. We are looking for the number of ways to choose four tires out of five, which is a permutation of 5 items taken 4 at a time (5P4).

Step 5: Calculating Arrangements with the Spare Tire

To calculate 5P4, we use the permutation formula n! / (n - k)! where n is the total number of items and k is the number of items to choose.

Step 6: Executing the Calculation for 5P4

Compute 5! / (5 - 4)!, which simplifies to 5! / 1!, since 1! is 1. Then calculate 5! which is 5 x 4 x 3 x 2 x 1.

Key Concepts

Factorial

A factorial, denoted as n!, represents the product of all positive integers from 1 to a given number n. It's a fundamental concept in the field of permutations and combinations as it helps in calculating the number of ways objects can be arranged.

For instance, if you have n objects and you want to know in how many different ways you can arrange them, you would calculate n!. Consider n to be 4; then 4! represents 4 times 3 times 2 times 1, which equals 24. This means there are 24 unique ways to arrange four distinct objects.

The factorial value grows extremely quickly with the increase of n, making it an important concept for handling larger sets of objects in probability and statistics as well.

Arrangements of Objects

The arrangement of objects refers to the different ways objects can be ordered or organized. When the order of items is important, such as in a sequence or pattern, we're dealing with permutations. In the case of the automotive example, arranging four tires on a car is a straightforward permutation task, as each tire takes a distinct position.

To illustrate, imagine labeling the tires A, B, C, and D. One possible arrangement is ABCD, but if we change the order to ACBD, we then have a completely different arrangement. Although the same four tires are used, their order changes the arrangement, embodying the essence of permutations in practical situations.

The concept of arrangements is pivotal in various fields, from seating guests at a wedding to assigning tasks in project management, highlighting its versatility and importance in problem-solving.

Permutation Formula

The permutation formula is used to calculate the number of possible arrangements (permutations) of a subset of items within a larger set. It's written in general form as nPr, where n is the total number of items and r is the number of items to be arranged. The formula for this is \( n! / (n - r)! \).

In the context of the tire arrangement problem, when including the spare tire, we're not arranging all five tires (n=5), but only selecting four of them (r=4). Here the formula becomes \( 5P4 \) which is calculated by 5! divided by (5-4)!, simplifying to 5!.

This formula handles situations where you have a large set but only want to focus on a specific number of items—a common scenario in probabilistic modeling, game theory, and resource allocation problems. Understanding this formula is essential for determining accurate permutations quickly and effectively.