Problem 7 - College Algebra

Question

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{8} P_{0} $$

Step-by-Step Solution

Answer

The solution to the expression \(_8 P_{0}\) is 1.

Step 1: Identify the values of n and r

In the expression \(_8 P_{0}\), \(n\) is 8 and \(r\) is 0.

Step 2: Apply the permutation formula

The general formula to calculate permutation is \(_n P_r = \frac{n!}{(n-r)!}\). But since \(r=0\), any number to the power 0 is equal to 1. Hence, \(_8 P_{0} = 1\

Step 3: Simplification

So the expression \(_8 P_{0} = 1\

Key Concepts

Factorials

When dealing with permutations and combinations, factorials play a crucial role. A factorial, denoted by the exclamation mark "!", is the product of all positive integers up to a specified number. For example, the factorial of 5, represented as 5!, is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are instrumental in finding the number of ways to arrange a set of objects. It simplifies counting and calculating probabilities in various scenarios. The concept of factorial is fundamental when applying permutation formulas, as these formulas often involve terms like \(n!\) and \((n-r)!\).
An important point to note is that the factorial of 0 is defined as 1, i.e., 0! = 1. This might seem counterintuitive at first, but it ensures consistency in mathematical formulas, particularly in combinatorial mathematics, where certain permutations or combinations involve choosing none (like in the case of \(^8 P_0\)).

Combinatorial Mathematics

Combinatorial mathematics is the branch of mathematics dealing with counting, arrangement, and combination of sets or numbers. It involves various techniques that help in finding methods of arranging different elements within a set.
Combinatorial problems are often about figuring out the number of ways to group or order certain elements. This could apply to real-world problems like determining seating arrangements at a dinner party, or efficiently managing tasks within a work setting.

• Permutation: This is a key aspect of combinatorics and refers to the arrangement of all or part of a set of objects, with regard to the order of the arrangement. For instance, how different letters can be arranged to form words.
• Combination: Unlike permutations, combinations are selections of objects where the order does not matter. For example, picking team members from a group.

Studying these techniques forms the basis for solving complex problems in fields ranging from computer science to probability theory. It provides tools to understand and solve practical questions where counting efficiently is essential.

nPr Formula

The \(nPr\) formula is an essential concept in permutations, used to calculate the number of ways to arrange \(r\) objects from a total of \(n\) objects, where order matters. The formula is expressed as:

• \(_n P_r = \frac{n!}{(n-r)!}\)

Here, \(n!\) represents the factorial of \(n\), which is the product of all positive integers up to \(n\). Meanwhile, \((n-r)!\) represents the factorial of the difference between \(n\) and \(r\). This subtraction accounts for objects that are not being selected or ordered.
The formula enables us to calculate situations where arrangement is important, like password creation, race rankings, or organizing library books.
In exercises like \(_8 P_0\), where \(r = 0\), the formula simplifies as the denominator, \((n-0)!\), equals \(n!\), leading to a result of 1. This addresses the notion that there's only one way to "arrange" zero things, no matter how many are available to choose from.