Problem 17 - College Algebra

Question

For the following exercises, compute the value of the expression. $$ P(3,3) $$

Step-by-Step Solution

Answer

The value of \( P(3,3) \) is 6.

Step 1: Understand the Permutation Notation

The notation \( P(n, r) \) represents the number of permutations of \( n \) items taken \( r \) at a time. This is used when the order of selection matters.

Step 2: Apply the Permutation Formula

The formula for permutations is given by:\[ P(n, r) = \frac{n!}{(n-r)!} \]Where \( n! \) is the factorial of \( n \), and \( (n-r)! \) is the factorial of \( n-r \).

Step 3: Substitute Values into the Formula

Given the problem, \( n = 3 \) and \( r = 3 \). Substitute these values into the permutation formula:\[ P(3, 3) = \frac{3!}{(3-3)!} \]

Step 4: Calculate Factorials

First, calculate \( 3! \) which is:\[ 3! = 3 \times 2 \times 1 = 6 \]Then, calculate \( 0! \), which by definition is:\[ 0! = 1 \]

Step 5: Compute the Permutation Value

Substitute the factorial values back into the formula:\[ P(3, 3) = \frac{6}{1} = 6 \]

Step 6: Interpret the Result

Thus, the number of ways to arrange 3 items out of 3 is 6.

Key Concepts

Factorials

In mathematics, a factorial is a specific kind of mathematical operation, which is very important in permutation calculations. To find the factorial of a non-negative integer, you multiply that integer by all the positive integers less than it. The notation for factorial is represented by an exclamation point (!). For example, if you have a number 3, its factorial is written as 3!, and can be calculated as follows:

• Start with the number itself (3)
• Multiply by 2
• Then multiply by 1

Therefore, we get the value: \[ 3! = 3 \times 2 \times 1 = 6 \]Factorials grow really quickly, meaning as the number increases, the factorial gets large fast! That is why it is crucial to understand how to calculate them, especially in permutation problems. Another important thing to note is that 0! is defined as 1. This might seem strange at first, but it is a useful definition and helps in simplifying expressions in permutations and combinations.

Permutation Formula

Permutations are all about arranging things, and they are key when you want to know how many different ways you can order a certain number of items. The formula to calculate permutations is represented as \( P(n, r) \). It helps calculate the number of ways to choose and arrange \( r \) things from \( n \) total things. The formula itself is:\[ P(n, r) = \frac{n!}{(n-r)!} \]Where \( n \) is the total number of items, and \( r \) is the number of items you are choosing to arrange. Here are a few points about permutations:

• Order Matters: Unlike combinations, permutations care about the order in which you arrange the items.
• Uses Factorials: As seen in the formula, calculations rely heavily on factorials for determining permutations.

By substituting in values of \( n \) and \( r \), you can solve permutation problems. For example, if \( n = 3 \) and \( r = 3 \), the calculation is simply evaluating \( \frac{3!}{(3-3)!} \), which results in 6 different ways to arrange the items.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. They help represent real-world problems in a form that can be manipulated mathematically. When dealing with permutations, understanding algebraic expressions is vital because permutation formulas are algebraic in nature.In the given permutation problem \( P(n, r) \), you replace \( n \) and \( r \) with numbers to create an algebraic expression which can be simplified using factorials.

• Substitute Values: Replace variables like \( n \) and \( r \) with given numbers to begin computation.
• Simplify Expressions: Use factorials to simplify and find the result.

For instance, in the exercise \( P(3, 3) \), you substitute and simplify using the permutation formula. Understanding how expressions change and develop by substitution and simplification is crucial when working through permutation and factorial-based problems.