Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ _{5} P_{2} $$

Step-by-Step Solution

Answer

The value of \(_{5}P_{2}\) is 20.

Step 1: Understand the Notation

The expression \( _{5}P_{2} \) is the notation for permutations. It represents the number of ways to arrange 2 items out of 5. The formula for permutations is \( _{n}P_{r} = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items and \( r \) is the number of items to arrange.

Step 2: Identify the Parameters

In the expression \( _{5}P_{2} \), the number of total items \( n \) is 5 and the number of items to arrange \( r \) is 2. Thus, we'll use \( n = 5 \) and \( r = 2 \) in our formula.

Step 3: Apply the Permutation Formula

Plug in the values into the permutation formula: \( _{5}P_{2} = \frac{5!}{(5-2)!} = \frac{5!}{3!} \).

Step 4: Calculate Factorials

Calculate the factorials. - \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)- \(3! = 3 \times 2 \times 1 = 6\)

Step 5: Simplify the Expression

Now divide the factorials: \[ _{5}P_{2} = \frac{120}{6} = 20 \].

Key Concepts

Understanding Factorials

To dive into the concept of permutations, it is essential to first understand what factorials are. A factorial, commonly denoted by an exclamation mark ("!"), is a product of all positive integers up to a given number. For instance, 5 factorial, written as \(5!\), is calculated by multiplying 5 by each of the numbers below it: \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

This idea of multiplying down to one is crucial when dealing with permutations and allows us to calculate the number of ways items can be arranged. Let's say we have \(n\) items, then \(n!\) gives us the number of ways to arrange all of them in a sequence. Understanding this will be key to applying the permutation formula effectively.

When dealing with permutation problems, part of what we often need to calculate involves factorials. It's like having a toolbox of numbers from 1 up to your total number (\(n\)) and seeing how these tools multiply to provide solutions in permutation problems.

Using the Permutation Formula

The permutation formula is a fundamental part of solving arrangement problems where order matters. It's expressed as \(_{n}P_{r} = \frac{n!}{(n-r)!}\). Here, \(n\) is the total number of items we have available, and \(r\) is how many of these we want to arrange.

Let’s discuss how the formula works by exploring each component:

• The numerator, \(n!\), represents the factorial of the total items, providing the total arrangements for \(n\) items.
• The denominator, \((n-r)!\), adjusts this by dividing through the factorial of the items we don't want to arrange, effectively zeroing out the arrangements that extend beyond \(r\) items.

So, if you need to find \(_{5}P_{2}\), it's saying "how many different arrangements can we make with 2 items out of 5?" As seen in the exercise, \(_{5}P_{2} = \frac{5!}{3!} = \frac{120}{6} = 20\).

Each specific arrangement of the 2 items from the set of 5 is unique due to the order, showcasing the importance of permutations in determining sequence-based solutions.

Solving Arrangement Problems

Arrangement problems focus on the sequence in which objects or numbers appear. These problems are answered using permutations, where each sequence is unique depending on order.

When tackling these problems, consider:

• How many items can be chosen or need to be arranged?
• Does the order in which you arrange these items matter? (Permutations are used when it matters!)
• What are the constraints for your selection?

An example is wanting to form a team of 2 players from 5 available players and knowing how many different ways this is possible. Using our formula \(_{n}P_{r}\), you quickly solve it with logic and calculation speed given by factorials.

Overall, these problems tend to appear in both simple daily settings and complex mathematical scenarios, making understanding of permutations a powerful tool for structuring and interpreting data where arrangement significance is prioritized.