Problem 43 - Algebra 2

Question

How many ways can eight runners in an Olympic race finish in first, second, and third places? F 8 G 24 H 56 J 336

Step-by-Step Solution

Answer

336 ways.

Step 1: Understand the Problem

We need to determine the number of ways to select the first three positions out of eight runners in a race. This is a permutation problem because the order in which the runners finish is important.

Step 2: Apply Permutation Formula

The formula for permutation is given by \( nPr = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to arrange. Here, \( n = 8 \) (runners) and \( r = 3 \) (positions: first, second, third).

Step 3: Calculate the Permutation

Using the permutation formula: \[ 8P3 = \frac{8!}{(8-3)!} = \frac{8 \times 7 \times 6 \times 5!}{5!} = 8 \times 7 \times 6 \].

Step 4: Simplify the Calculation

Simplify the expression: 8 multiplied by 7 multiplied by 6 gives 336.

Key Concepts

Permutation formula

In mathematics, the permutation formula is used to find the number of ways to arrange a specific number of items from a larger set, where the order of arrangement matters. This formula is commonly denoted as \( nPr \), and it is calculated using the equation \( nPr = \frac{n!}{(n-r)!} \). In this formula, \( n \) stands for the total number of items you have, and \( r \) represents the number of items you want to arrange.
To understand why order is crucial, let's consider a simple example. Imagine you have three colored balls: red, blue, and green. If you were to arrange all of them in different orders, you would have these sequences: red-blue-green, red-green-blue, blue-red-green, blue-green-red, green-red-blue, and green-blue-red. That's six different arrangements, showing that changing the order changes the permutation. In our specific exercise, we apply the permutation formula to arrange 3 positions (first, second, and third) from 8 runners. Thus, we use \( 8P3 \) to find our solution.

• Useful for problems where the order of selection matters.
• Common in scenarios like races where position order is crucial.
• The formula helps simplify and solve problems related to arrangement of objects.

Factorials

Factorials are a mathematical operation represented by an exclamation mark \(!\). The factorial of a non-negative integer \( n \), written as \( n! \), is the product of all positive integers from 1 to \( n \). For instance, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
Factorials appear in various fields of mathematics, most notably in permutations and combinations. They help calculate how many ways you can arrange a set of objects. In permutations, factorials are used to simplify the calculation of permutations by providing an easy-to-follow method for multiplying sequences of numbers.

• \( 0! \) is a special case, defined as 1.
• They grow rapidly, with larger numbers resulting in extremely large values.
• Integral in calculations involving permutations and combinations.

In the given exercise, we calculate \( 8! \) and \( 5! \) when determining the number of permutations for the top three positions among eight runners. This process involves simplifying the fraction by canceling common factorial terms.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of objects. It provides tools to solve problems like finding the number of possible configurations or selections in a given situation. In essence, combinatorics helps us understand how finite sets of elements can be arranged or selected.

There are two main types of combinatorial problems: permutations and combinations.

• Permutations concern the arrangement of items where the order matters, like assigning rankings to runners in a race.
• Combinations deal with selecting items where the order does not matter, such as choosing students to form a group.

In the process of solving the original problem of arranging three finishers out of eight runners, we specifically apply permutation principles. Combinatorics is essential not only for solving complex mathematical problems but also plays a crucial role in fields such as computer science, statistics, and various areas of engineering.