Problem 3 - 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. \(r=4, n=4\)

Step-by-Step Solution

Answer

There are 24 ordered selections of 4 objects from 4 objects.

Step 1: Understanding the Definition

We need to determine the number of ordered selections of \( r \) objects from a total of \( n \) objects. This scenario is described by permutations.

Step 2: Recall the Permutation Formula

The formula for permutations when selecting \( r \) objects from \( n \) objects is given by:\[P(n, r) = \frac{n!}{(n-r)!}\]where \( ! \) denotes a factorial.

Step 3: Substitute Known Values into the Formula

We know that \( r = 4 \) and \( n = 4 \). Substitute these values into the permutation formula:\[P(4, 4) = \frac{4!}{(4-4)!}\]which simplifies to \( P(4, 4) = \frac{4!}{0!} \).

Step 4: Calculate the Factorials

Calculate the factorials needed:- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)- \( 0! = 1 \) (a special mathematical definition).

Step 5: Compute the Permutation

Substitute the factorial values back into the formula:\[P(4, 4) = \frac{24}{1} = 24\]This means there are 24 different ways to select and order 4 objects from a collection of 4 objects.

Key Concepts

Factorial

Factorials are a cornerstone of permutations. When calculating permutations, you’ll often see the exclamation mark symbol (like this: \( n! \)). But what does it really mean? Essentially, a factorial is the product of all positive integers up to a given number. For example, \( 4! \) is calculated as \( 4 \times 3 \times 2 \times 1 \), which equals 24. Factorials grow rapidly. For instance, \( 5! = 120 \) and \( 6! = 720 \). This reflects the many possible ways to arrange a given set of items. Of special note is \( 0! \), which equals 1. It might seem odd at first, but this special case is crucial for making various formulas, like permutations, work smoothly without exceptions. When you think about it, \( n! \) is about counting all possible sequences that can be formed with \( n \) different objects. That's why it's widely used in problems involving order and arrangement.

Ordered Selections

Ordered selections, or permutations, refer to the different ways you can arrange a subset of items from a larger group. It's important to remember that order matters here. For instance, selecting a trio ABC is different from selecting BAC or CAB. Consider a small collection of items: suppose you have three colored balls—red, blue, and green. If you're asked how many unique sequences you can create by arranging these 3 balls, you'd think about permutations. In this example, you'd have the following sequences:

• Red, Blue, Green
• Red, Green, Blue
• Blue, Red, Green
• Blue, Green, Red
• Green, Red, Blue
• Green, Blue, Red

This shows every possible ordered arrangement when each item is distinct. Ordered selections help to solve many problems where distinguishing between combinations and permutations is necessary.

Permutation Formula

The permutation formula is your go-to tool for finding the number of ordered selections of items. When you hear permutations, think about arranging items where order is crucial. The formula for permutations is written as:\[ P(n, r) = \frac{n!}{(n-r)!} \]Here's what the notation means:

• \( n \) is the total number of items available.
• \( r \) is the number of items you've selected from that group.

This formula gives you the count of possible arrangements when selecting \( r \) objects from \( n \) without repeating any items.If \( n = 4 \) and \( r = 4 \), then by substituting these values into the formula, \( P(4, 4) = \frac{4!}{0!} = 24 \). This means there are 24 ways to arrange 4 items from a group of 4, considering every potential order.Understanding this formula helps in many scenarios, from simple rankings, to understanding complex data arrangements, where the sequence of items can drastically alter their meaning or outcome. It emphasizes the importance of order in certain mathematical situations.