Problem 29 - College Algebra

Question

Use the Fundamental Counting Principle to solve Exercises \(21-32\). Six performers are to present their comedy acts on a 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

There are 120 different ways to schedule the appearances.

Step 1: Decide on the Number of Positions

Given that there are 6 performers and one insists on performing last. This reduces the number of performers that we need to arrange in the first 5 positions to 5. This performer will automatically take the last, or 6th, position, so we don't have to calculate possible variations for that slot.

Step 2: Calculate the number of possible schedules for the first 5 performers

As each performer can only perform once, the number of ways to schedule the performers decreases with each scheduled performer. We start with 5 possible performers for the first slot, then 4 for the next, then 3, then 2, then 1. Therefore, the total number of schedules, according to the Fundamental Counting Principle, is the product of these possibilities: \(5 * 4 * 3 * 2 * 1 = 120 \).

Step 3: Include the performer who insists on performing last

This performer will always be in the last slot, so there is only 1 possibility for their placement. According to the Fundamental Counting Principle, we need to multiply this by the number of schedules for the first 5 performers: \(120 * 1 = 120\).

Key Concepts

Permutation

Understanding the concept of permutation is crucial when solving problems that involve arranging or ordering different items. In mathematics, a permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, if we have three objects A, B, and C, they can be arranged in several ways: ABC, ACB, BAC, BCA, CAB, and CBA. Each of these arrangements is a different permutation.

In the context of the exercise provided, we are dealing with a permutation scenario because the order in which the performers appear is significant. As there are 5 performers (excluding the one that performs last), we are interested in the permutations of the 5 performers. The number of permutations is calculated by multiplying the counts for each position available, which, in this case, starts with 5 options for the first position, 4 for the second, and so forth, finally reaching 1 for the last position. This gives us a total of 5 factorial ((5!) = 5 * 4 * 3 * 2 * 1 = 120) different permutations.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, such as algebra, probability, and geometry, and has applications in fields as diverse as computer science and statistical physics.

The example exercise falls under the umbrella of combinatorial theory because it involves counting the different ways of arranging the six performers. Specifically, we are applying combinatorial principles to calculate the number of possible arrangements or sequences in which acts could perform. Part of combinatorics, and particularly relevant to this exercise, is the use of the Fundamental Counting Principle to systematically count the number of permutations possible without having to list them all explicitly.

Arrangement of Performers

When working with arrangements such as scheduling performers, it's important to consider the conditions and constraints imposed by the situation. In our exercise, one performer insists on being the last, which is a restriction that influences the possible arrangements. Without such constraints, the total number of arrangements of six performers would be the factorial of six ((6!) = 720). However, with the constraint that one performer must go last, we exclude this position from the arrangement count.

This scenario is an example of how a simple change can significantly restrict the number of possible outcomes. With the last position fixed, we only need to arrange the remaining performers. It is also a perfect illustration of how the arrangement of performers is a real-life application of permutation concepts where order matters and constraints play a pivotal role in the calculation.

Counting Principle Exercises

The Fundamental Counting Principle is the guiding rule for solving various counting principle exercises. It allows us to find the number of possible outcomes without listing all the options. The principle states that if one event can occur in 'm' ways and another independent event can occur in 'n' ways, then the total number of combined occurrences or arrangements of these events is 'm * n'.

To reinforce student understanding, counting principle exercises often include multiple steps that break down the problem into smaller, more manageable parts. Applying this method to our exercise, we calculated possible schedules for the first five performers separately, then included the predetermined last performer to find the total number of different ways to schedule the acts. By practicing a variety of counting principle problems, students can become adept at using the principle to strategically and efficiently solve complex combinatorial problems.