Problem 27 - Algebra and Trigonometry

Question

19–32 These problems involve permutations. Running a Race In how many different ways can a race with nine runners be completed, assuming there is no tie?

Step-by-Step Solution

Answer

There are 362,880 different ways.

Step 1: Understand Permutations

Permutations are arrangements or orders of items. Here, the task is to determine in how many different orders nine runners can finish a race. Since there are no ties, each runner finishes in a unique position.

Step 2: Apply Factorial Formula

The number of permutations of nine distinct objects (runners) is calculated by finding the factorial of nine, denoted as \(9!\). This is because each of the nine runners could finish in any of the positions.

Step 3: Calculate 9!

Calculate \(9!\) as follows: \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362,880\).

Key Concepts

Factorial Formula

When you hear the term "factorial," it simply refers to the product of all whole numbers from 1 up to that number. The factorial of a number is a fundamental concept in permutations, which helps us find the total number of ways to arrange distinct objects. It is expressed with an exclamation mark (!). For example, for a given number "n," the factorial of "n" is denoted as \( n! \). This means multiplying \( n \) by every whole number less than itself, all the way down to 1.

When calculating permutations, the factorial formula makes it possible to determine the number of different arrangements possible. For example, if you want to find how many ways nine runners can finish a race, you would calculate \( 9! \) because each runner has a distinct finishing position.

Distinct Objects

Distinct objects are a crucial concept when determining permutations. "Distinct" means that each object is different from all the others in some way. In the context of a race, each runner is considered a distinct object because they all have unique characteristics and identities.

Using distinct objects is vital when applying permutation formulas, like the factorial formula, as it ensures each arrangement is unique. This is directly applicable to our runner scenario: each runner is different, and no two can occupy the same placement in a race, making the sequence of finishers distinctive.

If objects were not distinct, the issue of identical permutations would arise, as swapped positions would result in indistinguishable arrangements.

Order Arrangements

Permutations focus on the arrangement order of items. Order arrangements mean that the position of each object matters. The sequence in which objects appear changes the overall arrangement.

In the race problem, each finishing order is unique due to the importance of positioning; therefore, order arrangements help determine the outcome's possible sequences. Imagine lining up nine runners: the first runs only can finish in one spot, after that the choice for each next runner reduces by one until all have a finishing place.

This concept showcases the significance of order in permutations, distinguishing it from combinations where order does not matter.