Problem 37 - College Algebra

Question

Use the Fundamental Counting Principle to solve Six performers are to present their comedy acts on weekend evening at a comedy club. One of the performers insists on being the last stand-up comic of the evening. If this performer's request is granted, how many different ways are there to schedule the appearances?

Step-by-Step Solution

Answer

So, there are 120 different ways to schedule the appearances of the performers.

Step 1: Number of Performers

There are total 6 performers. One of the performers insists on being the last stand-up comic of the evening, that leaves 5 performers whose act order can be changed.

Step 2: Arranging the Remaining Performers

Except for the comic who wants to perform last, there are 5 performers left. There can be 5! (factorial) ways to arrange these performers. That is, first performer can be any of the 5, then the next one can be any of the remaining 4, then any of the remaining 3 and so on. Hence, it's a total of \(5*4*3*2*1\) which is 120.

Step 3: Scheduling the Last Performer

There is only 1 arrangement for the comic who wants to perform last. He can only be in the 6th or last place.

Step 4: Applying the Fundamental Counting Principle

According to Fundamental Counting Principle, total number of ways to arrange the 6 performers is the product of number of ways to arrange the first 5 performers and the number of ways to arrange the last performer, that is, \(120*1 = 120\)

Key Concepts

Counting Methods

Counting methods are essential tools in combinatorics that help determine the total number of ways events can happen. In the exercise, the problem revolves around finding how to schedule a group of comedians for performances. By using counting methods, we identify the different sequences in which performers can go on stage.

Key points to remember include:

• Breaking down the problem: Focus on smaller parts, like arranging a subset of performers.
• Using systematic counting strategies: In our case, the last performer is fixed, helping simplify the arrangement of others.

The critical idea here is recognizing constraints and finding logical ways to count arrangements without missing possibilities.

Factorials

Factorials are a key element when dealing with arrangements in counting problems. The factorial of a number, represented as \(n!\), is the product of all positive integers up to \(n\).

For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). This is crucial when considering different orders or sequences.

• It simplifies complex counting problems by reducing them to basic arithmetic expressions.
• Factorials describe the total number of ways to arrange \(n\) distinct objects.

Understanding factorials allows us to quickly compute the number of arrangements, especially in problems like scheduling and permutations.

Combinatorics

Combinatorics is the area of mathematics focused on counting, arranging, and combination of objects. It's applicable in many real-life scenarios like event planning and scheduling.

• Combinatorial reasoning is about making systematic choices.
• Fundamental Counting Principle (FCP) is a combinatorial tool used to calculate possible event sequences.

In the given exercise, FCP is used because it allows for multiplying the number of ways to arrange groups or steps to find the overall configurations. With the last comedian fixed, the FCP provides a straightforward path to calculate the remaining arrangements, demonstrating its practical use in combinatorial problems.