Question

A survey asks the respondent to order by importance 10 properties of a car. How many orderings are possible? How many orderings are there if the first and last property are given?

Short answer

There are 3,628,800 total orderings and 40,320 if the first and last properties are fixed.

Step-by-step solution

Understanding the Problem

We have 10 properties of a car, and we need to determine how many different ways they can be ordered. Furthermore, if the first and last properties are fixed, we need to find out how many orderings remain possible for the properties in between.

Total Number of Orderings

To find the total number of orderings of 10 properties, we calculate the factorial of 10. The formula for this is \(10!\).

Calculating 10!

Calculating \(10!\) involves multiplying the numbers from 1 through 10: \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800\). So, there are 3,628,800 different orderings possible.

Fixing the First and Last Property

When the first and the last properties are fixed, it leaves only the 8 properties in between to be ordered freely. Therefore, we calculate the number of orderings of these 8 properties, which is \(8!\).

Calculating 8!

Calculate \(8!\) by multiplying the numbers from 1 through 8: \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\). Therefore, there are 40,320 possible orderings if the first and last properties are fixed.

Key concepts

Factorial

The concept of a factorial is essential in combinatorics, particularly when calculating permutations. When you take the factorial of a number, you are multiplying that number by every positive integer less than it. For the number 10, this means multiplying all whole numbers down to 1:

• \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)
• This gives a product of 3,628,800.

The factorial function grows rapidly with larger numbers. For instance, the factorial of 8, denoted as \(8!\), involves a similar process and results in 40,320. Factorials are key in determining how many ways you can arrange a set of items. The formula for finding factorials is denoted by an exclamation mark, \(!\), following the number.

Permutations

Permutations are all about arrangements. When you have a set of items and you want to know how many ways you can order them, you're dealing with permutations. For example, if you have 10 properties of a car and want to find out how many different sequences you can create, you're looking at a permutation problem.

• Without any restrictions, the number of permutations of 10 items is \(10!\).
• This gives 3,628,800 different arrangements.

Now, if the first and last items are specified, it's a slightly different problem where only 8 items need arranging. This changes the permutation calculation to \(8!\), resulting in 40,320 ways to arrange the rest. Permutations consider the sequence as important; changing the order creates a different permutation.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values. In the realm of combinatorics, which is a fundamental area of discrete mathematics, topics like permutations and factorials play a major role. Each operation in this field considers finite or countable structures, like the finite set of 10 properties in a permutation problem.
Unlike continuous mathematics, which deals with values within a continuous range, discrete mathematics navigates through structures that aren't connected. It's pivotal for understanding scenarios where order and arrangement matter, such as determining possible orderings of items in a survey.

• Combinatorics, a branch of discrete math, helps us calculate possibilities and arrangements, including both permutations and combinations.
• It particularly shines when solving problems that involve counting specific arrangements or selections from a set.