Problem 29 - Precalculus with Limits

Question

In Exercises 29 - 34, evaluate \( _nP_r \). \( _4P_4 \)

Step-by-Step Solution

Answer

_4P_4 = 24

Step 1: Understanding the permutation formula

First understand the permutation formula: \( _nP_r = \frac{n!}{(n-r)!} \). This formula is used to calculate the number of arrangements of n items taken r at a time.

Step 2: Plugging the values into the formula

Next, plug the given values into the permutation formula: \( _n P_r = \frac{n!}{(n-r)!} \) to get \( _4 P_4 = \frac{4!}{(4-4)!} \)

Step 3: Computing the permutation

Now calculate the permutation. Firstly, calculate the factorial of 4, which is 4! = 4 * 3 * 2 * 1 = 24. Secondly compute the factorial of (4-4), which is 0! = 1(as by definition, the factorial of 0 is 1). Finally, divide the factorial of 4 by the factorial of (4-4) to get the permutation _4P_4 = 24.

Key Concepts

Permutation Formula

Understanding the permutation formula is critical for solving problems related to the arrangement of items. The permutation formula is represented as \( _nP_r = \frac{n!}{(n-r)!} \).

This formula is used when we want to determine the number of possible ways to arrange or order \( r \) items out of a pool of \( n \) distinct items where the order does matter. It becomes particularly useful for problems where you're dealing with ordered sequences without any repetition of items.

To break it down:

• \( n \) represents the total number of items you can choose from.
• \( r \) signifies the number of items you want to arrange.

When calculating permutations, remember that different orders count as different arrangements even if the objects are the same.

Factorial Notation

Factorial notation plays a foundational role when working with permutations. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \).

For a clear example, here’s how you would calculate the factorial of 5:

• \( 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 \).

Moreover, it’s important to be aware of the special case where \( 0! = 1 \), even though there are no positive numbers to multiply. This concept ensures that permutation expressions work correctly even when we are choosing zero items from a set, as seen in the provided exercise.

Arrangements of Items

The concept of arrangements centers around the question: 'In how many different ways can we arrange a subset of items from a larger set?'. For instance, if you have four distinct books, you might ask how many different ways can you arrange them on a shelf.

When using permutations to solve problems about arrangements, it’s essential to consider two main factors: the total number of items \( n \) available, and how many of those you’re trying to arrange, \( r \). The bigger the values of \( n \) and \( r \), the more complex the arrangement possibilities become. On the other hand, if \( r = n \), every item available is used in the arrangement, and this is often referred to as a 'full permutation'.

Combinatorics in Precalculus

Combinatorics is a branch of mathematics dealing with counting, both as a means and an end in obtaining results, and certain properties of finite structures. In precalculus, combinatorics lays the groundwork for understanding more complex probability and statistics topics later on.

Within the realm of combinatorics, permutations serve as a fundamental concept to understand various types of problems like those involving arranging items, distributions, and more. The ability to discern whether a problem is asking for a permutation or combination (where order does not matter) is crucial for finding the correct solution. If order is important, then permutations are the tool of choice for modeling these scenarios.