Problem 5 - Algebra 2 and Trigonometry

Question

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects without replacement. \(r=3, n=8\)

Step-by-Step Solution

Answer

The number of ordered selections is 336.

Step 1: Define the Problem

We need to find the number of ordered selections of 3 objects from a collection of 8 objects without replacement. This is a permutations problem, where the order of selection matters.

Step 2: Understand Permutations Formula

The number of permutations of selecting \(r\) objects from \(n\) objects is given by the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \]Here, \(n!\) denotes the factorial of \(n\) and \((n-r)!\) denotes the factorial of \(n-r\).

Step 3: Substitute the Values

Substitute \(n=8\) and \(r=3\) into the permutations formula:\[ P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} \]

Step 4: Compute the Factorials

Calculate the factorials: - \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) - \(5! = 5 \times 4 \times 3 \times 2 \times 1\)

Step 5: Simplify the Expression

Substitute the computed factorials into the expression:\[ \frac{8!}{5!} = \frac{8 \times 7 \times 6 \times 5!}{5!} \]Cancel \(5!\) in the numerator and denominator:\[ 8 \times 7 \times 6 = 336 \]

Step 6: Conclusion

The number of ordered selections of 3 objects from a collection of 8 objects is 336.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially in permutations and combinations. The factorial of a number, represented by an exclamation mark "!", is the product of all positive integers up to that number. For example:

• The factorial of 5, noted as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
• Similarly, the factorial of 8, written as \(8!\), equals \(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\).

Factorials are essential for calculations involving permutations because they help in counting the number of ways to arrange objects. Remember, the factorial of zero, \(0!\), is defined as 1 by convention.

Ordered Selections

When we talk about ordered selections, we're referring to scenarios where not only the choice of items matters, but also the order in which you pick them. This can be crucial in areas like creating passwords or seating arrangements. Imagine you have a set of eight books, and you want to select three to arrange on a shelf. The order you place the books in will affect the overall look, making it an ordered selection problem. In mathematics, this concept is handled using permutations, which allow us to calculate the total number of possible ordered sequences. Using the permutation formula, you can determine exactly how many ways you can select and arrange items from a larger group.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and combinations of sets of elements. It helps us determine how to group, arrange, and select objects in a mathematical way. In real-world applications, combinatorics can solve everything from configuring network topology to game theory strategies. There are two primary aspects to combinatorics:

• Combinations: When the order doesn’t matter.
• Permutations: When the order does matter.

In our exercise, since choosing 3 items out of 8 in a specific order matters, we are dealing with permutations, making this a classic example of combinatorial math. Understanding these concepts is key to solving problems that involve any form of arrangement or selection.

Permutations Formula

The permutations formula is a tool used to calculate the number of ways to pick ordered selections. The formula looks like this: \[ P(n, r) = \frac{n!}{(n-r)!} \]Here:

• \(n\) is the total number of objects available to pick from.
• \(r\) is the number of objects you want to pick.
• \(n!\) is the factorial of \(n\), while \((n-r)!\) is the factorial of \((n - r)\).

This formula gives the number of different ways you can choose \(r\) objects from a larger set of \(n\) objects, with the order of selection being important.For example, in our exercise, choosing 3 objects from 8 is calculated as:\[ P(8, 3) = \frac{8!}{5!} = 8 \times 7 \times 6 = 336 \]The permutations formula is widely used in fields requiring combination and arrangement analysis, such as statistics, computer science, and logistics.