Problem 34 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(6,6)\)

Step-by-Step Solution

Answer

\( P(6, 6) = 720 \).

Step 1: Understand the Permutation Formula

The notation \( P(n, r) \) stands for the number of permutations of \( n \) items taken \( r \) at a time. The formula to calculate it is \( P(n, r) = \frac{n!}{(n-r)!} \). Here, we need to find \( P(6,6) \), which means we are finding the permutations of 6 items taken all 6 at a time.

Step 2: Substitute Values into the Formula

Substitute \( n = 6 \) and \( r = 6 \) into the permutation formula: \( P(6, 6) = \frac{6!}{(6-6)!} \).

Step 3: Calculate the Factorials

First, calculate the factorials: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \) and \( 0! = 1 \) since by definition the factorial of 0 is 1.

Step 4: Compute the Permutation Value

Using the calculated factorials, substitute back into the permutation expression: \( P(6, 6) = \frac{6!}{0!} = \frac{720}{1} = 720 \). The number of permutations of 6 items taken 6 at a time is 720.

Key Concepts

Factorials

A factorial of a non-negative integer is a key concept in permutations and combinations. If you wonder what the curious exclamation mark means in mathematical expressions, it indicates a factorial. Given a number, say 6, the factorial is represented as 6! and is calculated as the product of all positive integers up to that number.
For example, to calculate 6!, you multiply 6 by 5, by 4, down to 1. In formula terms:

• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

It’s also crucial to know that the factorial of 0 is defined as 1, by convention. This definition helps maintain the consistency of calculations in formulas like permutations and combinations. Factorials are foundational in these areas and simplify the computation when dealing with larger sets.

Permutation Formula

When dealing with permutations, we are essentially looking at ways to arrange a set of items. The notation \(P(n, r)\) represents the number of permutations of \(n\) items taken \(r\) at a time. This can be calculated using the permutation formula:

• \(P(n, r) = \frac{n!}{(n-r)!}\)

To understand the formula, consider how we have \(n!\) ways to arrange \(n\) items. By dividing by \((n-r)!\), we exclude arrangements that involve fewer than \(r\) items, hence "taking \(r\) at a time."
In the specific example of \(P(6,6)\), since we are taking all 6 items (\(r = n\)), the formula simplifies as all terms cancel except the numerator, resulting in just \(n!\), which here is 6! or 720. This tells us that there are 720 ways to arrange 6 items fully.

Evaluation of Expressions

Evaluating mathematical expressions involving permutations can initially appear complex, but breaking them into smaller steps simplifies the task. When we examine \(P(6, 6)\), this asks us to find how many ways we can arrange 6 items taken all 6 at a time. The steps include:

• Substituting the values into the permutation formula: \(P(6, 6) = \frac{6!}{0!}\).
• Calculating the factorials: We already know \(6! = 720\) and, by convention, \(0! = 1\).
• Carrying out the division \(\frac{720}{1} = 720\).

This process led us to the conclusion that there are 720 permutations for this specific case. Understanding how to evaluate similar expressions relies on a firm grasp of factorials and the permutation formula, allowing you to solve permutations for any given \(n\) and \(r\).