Problem 31 - Precalculus with Limits

Question

In Exercises 29 - 34, evaluate \( _nP_r \). \( _8P_3 \)

Step-by-Step Solution

Answer

The value of \( _8P_3 \) is 336.

Step 1: Understanding Permutation Formula

Firstly, understand the formula used for permutations, which is: \(_nP_r = \frac{n!}{(n-r)!}\) where 'n' is the total number of items, 'r' is the number of items to choose, and '!' denotes a factorial, i.e., the product of an integer and all the integers below it.

Step 2: Substitute Values

We substitute the given values into the permutation formula, \(P_8^3 = \frac{8!}{(8-3)!}\).

Step 3: Calculate Factorials

Now, calculate the factorials. The factorial of 8 is 40320 and the factorial of (8-3)=5 is 120.

Step 4: Evaluate The Equation

Evaluate the equation by dividing 40320 by 120 to determine the number of permutations, which yields 336.

Key Concepts

Factorial

When delving into permutations and combinations, one of the fundamental concepts to understand is the factorial. In mathematics, a factorial of a positive integer 'n', denoted by 'n!', is the product of all positive integers less than or equal to 'n'. For example, the factorial of 5, written as '5!', would be calculated as:
\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials grow rapidly with larger numbers, as seen with the number '8' in our exercise, where \( 8! = 40,320 \). In combinatorics, the factorial function is used extensively to calculate the number of ways in which objects can be arranged, which is the essence of counting permutations.

Combinatorics

Combinatorics is a field of mathematics focused on counting, but at its core, it's really about organizing and arrangement. It deals with combinations and permutations—how we can combine different sets of things, and in how many ways. Permutations are arrangements where the order does matter. Imagine you're picking a president, vice president, and secretary from a group of 8 people. The order you pick them in is important—that's a permutation. On the other hand, if you're forming a team of 3 people out of 8, and the order isn't important, that's a combination.
Applying combinatorics, students can solve a variety of problems, from simple arrangements to complex probability questions. In our textbook exercise, by understanding permutations, we can figure out the number of different ways to arrange a subset of a larger set, which is a key component of this mathematical area.

Permutation Formula

To take a deeper dive, the permutation formula, denoted as \( _nP_r \), helps us calculate the number of ways to arrange 'r' objects from a set of 'n' objects. The standard permutation formula is \( _nP_r = \frac{n!}{(n-r)!} \), where 'n' represents the total number of objects, and 'r' is the number of objects selected for arrangement.
To solve our given problem, \( _8P_3 \), we substituted the values into the permutation formula, yielding \( \frac{8!}{(8-3)!} \). This calculation involves factorials, as discussed previously. By simplifying the factorials, we can find that there are 336 different ways to arrange 3 objects from a set of 8.

Precalculus

Precalculus is a course that includes mathematical concepts required before diving into the world of calculus. It covers a range of topics that underpin calculus, including algebra, geometry, and yes, combinatorics. Understanding permutations and factorials is an essential part of precalculus, as it builds the foundational skills for evaluating complex mathematical problems.
Students often encounter this material as they prepare for calculus, gaining intuition for the behavior of mathematical functions, analyzing the warping of space, or pondering the chances of an event happening. The permutation exercise, like the one provided, serves as a fitting example of the practical application of precalculus concepts, helping students develop their problem-solving skills and enhance their mathematical toolset.