Problem 4 - College Algebra with Corequisite Support

Question

Answer the following questions. Describe how the permutation of \(n\) objects differs from the permutation of choosing \(r\) objects from a set of \(n\) objects. Include how each is calculated.

Step-by-Step Solution

Answer

Permutation of all n objects is \(n!\); choosing r from n is \(\frac{n!}{(n-r)!}\).

Step 1: Understanding Permutation of N Objects

A permutation of all n objects is an arrangement of all the objects in a specific order. The number of permutations of n objects is calculated using the factorial function, denoted as \(n!\). This is because every object can occupy every position: the first position can be filled by any of the n objects, the second by n-1, continuing this way until the last object is placed. Thus, the number of permutations of n objects is \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).

Step 2: Understanding Permutation of Choosing R Objects from N

The permutation of choosing r objects from a set of n objects is an arrangement of a subset of the objects in a specific order. This is calculated by first finding the permutations of the n objects (as we might choose any of the n objects), and then dividing by the permutations of the (n-r) objects that are not chosen, which results in \(\frac{n!}{(n-r)!}\). This accounts for selecting and ordering r objects from n.

Step 3: Comparing the Two

Permutations of n objects (\(n!\)) involve arranging all objects, whereas permutations of choosing r objects from n (\(\frac{n!}{(n-r)!}\)) involve choosing and arranging only r objects. The formula for selecting r objects considers that we don't want permutations of all n objects, but only the selected subset.

Key Concepts

Factorial function

The factorial function is a fundamental concept in combinatorics and mathematics.
It is often represented by the symbol "!" and is used to calculate the total number of possible arrangements of a set of objects. For any non-negative integer \(n\), the factorial \(n!\) is defined as:

• \(n! = n \times (n-1) \times (n-2) \times \, \ldots \, \times 1\).

Imagine you have a set of \(n\) unique objects. The factorial function computes how many different ways you can arrange all of these objects in a sequence.
For instance, if \(n = 3\), then \(3! = 3 \times 2 \times 1 = 6\).
This means there are six different ways to arrange three objects.
Understanding the factorial is essential for calculations in more complex permutations and combinations.
It lays the foundation for determining arrangements in various scenarios.

Arrangement of objects

Arranging objects in a specific order is what permutations are all about.
In this context, we often distinguish between arranging all the objects and selecting only a few to arrange.
The permutation formula helps us find this arrangement:

• For a full set of \(n\) objects, use \(n!\).
• For a subset of \(r\) objects chosen from \(n\) objects, use \(\frac{n!}{(n-r)!}\).

The crucial difference lies in whether you use all objects or just a part of them.

• If you arrange all \(n\) objects, each object has a unique position from one to \(n\).
• If you choose \(r\) objects from \(n\), you're not only selecting \(r\) objects but also ordering them specifically among themselves.".

Understanding these arrangements is crucial for solving problems related to permutations where order matters. It helps us analyze sequences efficiently, from seating arrangements to designing schedules.

Subset selection

Subset selection refers to choosing a specific number of objects from a larger set without considering the order.
This differs from permutations, where both selection and order matter.
In permutations, when selecting \(r\) objects from \(n\), we care about how we arrange them, but the initial step is always about choosing those \(r\) objects first.

• The formula \(\frac{n!}{(n-r)!}\) summarizes this by counting possible selections and arrangements.

While calculating it, you first account for the number of ways to fill \(r\) positions with \(n\) items.
The numerator \(n!\) represents the total arrangements of \(n\) items.
The denominator \((n-r)!\) cancels out the arrangements of the remaining \(n-r\) items.
Subset selection is the stepping stone to problems involving order, size, and grouping within larger statistical and probabilistic contexts.