Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Use the fundamental principle of counting or permutations to solve each problem. A concert to raise money for an economics prize is to consist of 5 works: 2 overtures, 2 sonatas, and a piano concerto. In how many ways can a program with these 5 works be arranged?

Step-by-Step Solution

Answer

There are 120 different ways to arrange the 5 works.

Step 1: Understanding the Problem

We need to find the number of ways to arrange 5 musical pieces, consisting of 2 overtures, 2 sonatas, and 1 piano concerto. The arrangement of these pieces is important, making it a permutation problem.

Step 2: Define the Elements

Identify the types of works: 2 overtures (O1, O2), 2 sonatas (S1, S2), and 1 piano concerto (C). We have a total of 5 works to arrange.

Step 3: Calculate the Total Arrangements

Since the total number of pieces is 5, and we need to arrange each work in the sequence, we calculate the total permutations as the factorial of 5. This is given by 5!, which is the product of all positive integers up to 5.

Step 4: Perform the Calculation

Compute 5! to find the total number of arrangements: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Step 5: Interpret the Result

There are 120 different ways to arrange the 5 works in the concert program. Each arrangement sequence is unique and accounts for the position of each musical piece.

Key Concepts

Fundamental Principle of Counting

The fundamental principle of counting is a core idea in mathematics. It's a method to determine the number of ways several tasks can be performed in sequence. This principle states that if there are multiple tasks, the total number of ways to perform all tasks is the product of the ways to perform each task, assuming the tasks are independent.

For example, if you need to choose an appetizer and a main dish for dinner, where there are 3 appetizers and 4 main dishes, the fundamental principle of counting tells us there are \(3 \times 4 = 12\) different meal combinations. This principle is incredibly helpful in a variety of situations where you need to count the possibilities without listing all combinations.

In the context of musical arrangements, this principle is used to calculate how many different ways you can arrange a series of musical works. It simplifies complex counting problems into manageable calculations by breaking them down into a series of simple multiplication steps.

Factorial

Factorial is a mathematical operation denoted by an exclamation mark (!). It's used to find the number of ways to arrange a set of items. Specifically, the factorial of a positive integer \(n\) is the product of all positive integers less than or equal to \(n\).

For example, \(5!\), which is read as "five factorial," is calculated as follows:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

This operation is significant in problems involving combinations and permutations. When you need to determine how many different ways you can arrange \(n\) items, calculating \(n!\) provides the answer instantly.

In the problem of arranging the musical works, using factorial helps find all possible sequences by computing \(5!\). This tells us there are 120 unique ways to arrange the 5 pieces in a line.

Musical Arrangements

Arranging music pieces in a concert program is a great way to apply the concept of permutations. When you have distinct pieces, the order in which they are played makes a difference, making permutations the perfect tool to use.

Unlike combinations, where order doesn't matter, permutations focus on every unique sequence possible. For a concert with 5 pieces, as in our exercise, each specific arrangement of these 5 works creates a new program.

Imagine having different instruments or genres: every order might convey a different impact on the audience. Therefore, the permutation counts each sequence as unique, much like how every performance can feel different to an audience even when playing the same pieces. This makes permutations valuable when organizing events or solving real-life problems involving order and arrangement.