Problem 33 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(8,1)\)

Step-by-Step Solution

Answer

The answer is 8.

Step 1: Understand the Permutation Notation

The notation \(P(n, r)\) represents the number of permutations of \(n\) things taken \(r\) at a time. In other words, it's about selecting \(r\) objects from \(n\) distinct objects and arranging them in a specific order.

Step 2: Apply the Permutation Formula

The formula for finding permutations is given by:\[P(n, r) = \frac{n!}{(n-r)!}\]For this exercise, we need to find \(P(8, 1)\). So we substitute \(n = 8\) and \(r = 1\) into the formula:

Step 3: Substitute Values into Formula

Substitute \(n = 8\) and \(r = 1\) into the permutation formula:\[P(8, 1) = \frac{8!}{(8-1)!} = \frac{8!}{7!}\]

Step 4: Simplify the Factorial Expression

Simplify \(\frac{8!}{7!}\):Remember that \(8! = 8 \times 7!\), so:\[\frac{8!}{7!} = \frac{8 \times 7!}{7!} = 8\]We've canceled out the \(7!\) terms.

Key Concepts

Factorial

The concept of the factorial, in mathematics, is a fundamental part of permutations and combinations. The notation for factorial is represented by an exclamation mark \(n!\), and it describes the product of all positive integers up to a certain number \( n \). For instance:

• \( n! = n \times (n-1) \times (n-2) \times \dots \times 1 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials are only defined for non-negative integers. By definition, \( 0! = 1 \), which might seem strange but works smoothly in calculations involving permutations and combinations. In the original exercise, the factorial is used in the formula for calculating permutations. Understanding factorial operations is essential for simplifying expressions like \( \frac{n!}{(n-r)!} \), which allows us to focus only on the required terms of the product.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In solving permutation problems like \( P(8,1) \), algebra helps us understand how expressions can be simplified through manipulation. Let's break it down with an example from our problem:

• We use the expression \( \frac{8!}{(8-1)!} \)
• This translates algebraically to \( \frac{8!}{7!} \)

Here, algebraic rules of simplifying expressions can be applied. Since \( 8! = 8 \times 7! \), we notice that cancelation is possible, which helps simplify the expression to 8. Algebraic methods are crucial for manipulating factorials and understanding the deeper structures within combinatoric problems.

Combinatorics

Combinatorics is the study of counting, arranging, and grouping items in specific ways. It's important in permutations where order matters. In the permutation \( P(8,1) \), combinatorics helps us determine how many ways we can arrange 1 object from a set of 8 distinct items. Using the formula:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

This formula is a powerful tool in combinatorics for calculating permutations. Combinatorics helps not only in counting arrangements but also in understanding different scenarios in probability and statistics. In essence, it empowers us to solve problems related to both small-scale scenarios and large-scale complex datasets where ordering of selection counts.