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Chapter 8: Problem 37

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?

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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, \(120*1 = 120\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It makes a difference whether or not I use parentheses around the expression following the summation symbol, because the value of \(\sum_{i=1}^{5}(i+7)\) is \(92,\) but the value of \(\sum_{i=1}^{8} i+7\) is $43 .

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller In how many ways can these six jokes be ranked from best to worst?

Exercises \(85-87\) will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots .\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}}\)and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Give an example of two events that are not mutually exclusive.

Use the formula for \(_{n} C_{r}\) to solve An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?
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