Problem 75 - Algebra 2

Question

Simplify each expression. $$ _{5} \mathrm{P}_{2} $$

Step-by-Step Solution

Answer

The simplification of \(_5P_2\) is 20.

Step 1: Identify n and r

From the given problem, \(_5P_2\), the total number of items \(n = 5\) and the number of items to arrange \(r = 2\).

Step 2: Apply the Permutation Formula

Now that we have identified our \(n\) and \(r\), we can apply the permutation formula \(P_{n,r} = \frac{n!}{(n-r)!}\). In this case, that is \(P_{5,2} = \frac{5!}{(5-2)!}\).

Step 3: Calculate Factorials and Simplify

Next, calculate the factorials and simplify the expression. 5 factorial means the product of all positive integers from 1 to 5, 5!=5 × 4 × 3 × 2 × 1 = 120. Also, (5 - 2)! is 3 factorial, 3!=3 × 2 × 1 = 6. Therefore, the expression becomes \(\frac{120}{6}\). Simplifying gives the result of 20.

Key Concepts

Factorials

When diving into permutations, the concept of factorials is a foundational one. A factorial, denoted as \(n!\), represents the product of all positive integers up to a given number \(n\). For instance, \(5!\) equals \(5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials are a useful mathematical tool that helps us understand arrangements and order in sets of numbers or objects.

Factorials grow quickly as \(n\) increases. For example:

• \(3! = 6\)
• \(4! = 24\)
• \(5! = 120\)

Factorials are especially important in permutations and combinations because they help calculate the different ways to arrange a set number of elements. Whenever we talk about arranging or ordering, factorials often come into play. Understanding this growth helps manage questions about order rearrangements in larger sets efficiently.

Permutation Formula

Permutations help us determine the number of ways to arrange a specific number of objects from a larger set. To compute permutations, we use a special formula that involves factorials. The permutation formula is given by \(P_{n,r} = \frac{n!}{(n-r)!}\).

Here, \(n\) represents the total number of items you have, while \(r\) indicates how many of those items you want to arrange. In the example of \(_5P_2\), \(n = 5\) and \(r = 2\), meaning we want to know how many different ways we can select and arrange 2 items out of 5.

Let's break down the process:

• Identify \(n\) and \(r\).
• Substitute them into the formula \(\frac{n!}{(n-r)!}\).
• Calculate the factorials \(n!\) (for the total items) and \((n-r)!\) (for the difference of items left after pairing).

After evaluating these factorials, you divide them to get the total number of permutations possible. This mathematical approach provides clarity in arrangements by focusing on how selection and order matter.

Combinatorics

Combinatorics is the branch of mathematics that studies finite or countable discrete structures. It involves counting, arrangement, and combination principles that play a vital role in various mathematical problems. Within combinatorics, permutations and combinations are two primary methods for considering how objects can be ordered or grouped.

Permutations, as we've learned, prioritize order, meaning different sequences count as different permutations. In contrast, combinations focus on groupings where order does not matter. This distinction illustrates the flexible tools that combinatorics offers to solve problems where the arrangement of elements is key.

Understanding permutations and combinations in combinatorics allows one to model and solve real-life scenarios ranging from scheduling, organizing tournaments, to cryptography. Knowing when to use permutations (where order is important) or combinations (where order is not) helps solve problems more intuitively and effectively.