Problem 61 - A Graphical Approach to College Algebra

Question

Use any or all of the methods described in this section to solve each problem. Course Schedule If Kim Falgout has 8 courses to choose from, how many ways can she arrange her schedule if she must pick 4 of them?

Step-by-Step Solution

Answer

Kim can arrange her schedule in 1680 different ways.

Step 1: Understanding the Problem

Kim needs to arrange 4 courses out of 8 available courses, and the order of these selected courses matters. This situation involves permutations since the arrangement is important.

Step 2: Choosing the Permutation Formula

Since the order matters, we will use the permutation formula which is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \]where \( n \) is the total number of courses to choose from (8), and \( r \) is the number of courses to select (4).

Step 3: Substituting Values

Plug the given values into the permutation formula:\[ P(8, 4) = \frac{8!}{(8-4)!} = \frac{8!}{4!} \]

Step 4: Calculating Factorials

Calculate \( 8! \) and \( 4! \): - \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \) - \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Step 5: Final Calculation

Substitute the factorial values into the formula:\[ P(8, 4) = \frac{40320}{24} = 1680 \]

Step 6: Concluding the Solution

Kim can arrange her schedule in 1680 different ways when selecting 4 courses out of the 8 available.

Key Concepts

Factorials

Factorials are a mathematical concept used in permutations and combinations. The factorial of a number, denoted by the symbol \( ! \), is the product of all positive integers less than or equal to that number.
For example, the factorial of 4 (\(4!\)) is the product of all numbers from 1 to 4:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Factorials grow very rapidly as the number increases. This property makes them useful for calculating permutations, where the arrangement of a set number of items is considered, and combinations, where only the selection is important without regard to order. In the context of the exercise, factorials help find the total number of possible arrangements of courses Kim can choose from.
To illustrate, \(8!\) represents the number of ways to arrange all 8 courses, and \((8-4)! = 4!\) represents removing 4 courses from consideration, reflecting the specific arrangement of the remaining ones.

Combinatorics

Combinatorics is the field of mathematics focused on counting, arrangement, and combination of elements within sets. It is particularly useful in determining the number of ways to arrange or choose objects.
In our exercise, permutations, a subfield of combinatorics, is applied as we deal with arranging 4 courses out of 8. Since order matters in this context, we use the permutation formula:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

Here, \( n \) is the total number of courses (8), and \( r \) is the number to arrange (4).
Using combinatorial methods, we get the total number of schedules Kim can arrange her courses in, taking into account every possible order. This approach is crucial not only in scheduling but also in broader fields like probability and optimization.

Course Selection

Course selection is an example of a real-world application of permutations and combinatorics. In Kim's situation, she has 8 available courses but needs to select and arrange 4 into a schedule.
The difficulty lies in how many ways she can line them up, where each choice impacts the arrangement of her entire schedule. Since the order in which courses appear is significant, permutations are the correct method to determine possible arrangements.
When solving problems like these, it's helpful to break it down into steps:

• Identify the total number of items (courses, in this instance)
• Determine how many items need arranging
• Choose the appropriate formula
• Calculate using factorials to find the total number of permutations

In Kim's case, the result of 1680 different arrangements highlights just how many potential schedules can be organized when the order of course selection is considered.