Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$P(5,5)$$

Step-by-Step Solution

Answer

\( P(5, 5) = 120 \)

Step 1: Understanding Permutations

The expression \( P(n, r) \) represents the number of ways to arrange \( r \) objects from \( n \) distinct objects. It is calculated using the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \] In our problem, \( n = 5 \) and \( r = 5 \).

Step 2: Calculate Factorials

First, we need to calculate \( 5! \), which is the factorial of 5. The factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \): \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Step 3: Apply the Permutation Formula

We plug \( n = 5 \) and \( r = 5 \) into the permutation formula: \[ P(5, 5) = \frac{5!}{(5-5)!} = \frac{5!}{0!} \]It's important to note that \( 0! = 1 \).

Step 4: Simplify the Expression

Now, simplify the expression using the calculated factorial values: \[ \frac{5!}{0!} = \frac{120}{1} = 120 \] Thus, \( P(5, 5) = 120 \).

Key Concepts

Factorials

Understanding factorials is crucial when dealing with permutations and combinations in mathematics. A factorial, represented by the exclamation mark "]!", is a product of all positive integers up to a given number. The notation for the factorial of a number \( n \) is \( n! \). For example, the factorial of 5, which can be written as \( 5! \), is the multiplication of numbers from 1 to 5:

• \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials are used extensively in permutations and combinations calculations because they represent the number of ways to arrange or select items. It's also essential to know that \( 0! = 1 \). This might seem strange since 0 isn’t a positive integer, but in mathematical convention, it allows for consistent calculations, especially within permutation and combination formulas.

Permutation formula

The permutation formula is a valuable tool for calculating the number of ways to arrange a specific number of items from a larger group. The formula for permutations is given by \( P(n, r) = \frac{n!}{(n-r)!} \), where:

• \( n \) represents the total number of items.
• \( r \) is the number of items to arrange.

For the exercise, the permutation \( P(5, 5) \) calculates how many ways you can arrange 5 items from a group of 5, which essentially means arranging all items.Applying the formula:

• \( P(5, 5) = \frac{5!}{(5-5)!} = \frac{5!}{0!} = \frac{120}{1} = 120 \)

Therefore, there are 120 ways to order 5 distinct items. This permutation formula becomes particularly useful in scenarios where the sequence matters, like arranging books on a shelf or scheduling events.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arranging, and combination of objects. It provides the fundamental toolset for solving problems related to permutations and combinations.In combinatorics, the distinction between permutations and combinations is paramount.

• Permutations: The order of items matters, and the permutations formula \( P(n, r) \) is used.
• Combinations: The order of items does not matter, and a different formula is applied.

Understanding these concepts helps in various real-world applications. For example, determining the number of potential outcomes in a sequence, such as possible passwords, combination locks, or different team lineups.In essence, combinatorics is about exploring all possible arrangements and selections, helping us solve complex counting problems efficiently. Hence, as seen in this exercise, these principles are applied to accurately compute specific scenarios like the number of permutations possible when arranging a set number of items.