Problem 38 - Algebra and Trigonometry

Question

Five singers are to perform at a night club. One of the singers insists on being the last performer of the evening. If this singer’s request is granted, how many different ways are there to schedule the appearances?

Step-by-Step Solution

Answer

There are 24 different ways to schedule the appearances.

Step 1: Determine number of singers who can be arranged

There are 4 singers (excluding the one who insists on performing last) who can be arranged to perform. The order in which these performers go matters, so this is a permutation.

Step 2: Calculate total possible permutations

The total number of possible permutations among 4 different performers can be calculated using the formula for permutations of n objects, which is n!. So the total permutations would be 4!.

Step 3: Evaluate the factorial

Evaluating the factorial of 4 gives a total of \(4*3*2*1 = 24\) different ways to arrange the first 4 performers.

Step 4: Fix the last performer

The last performer is fixed as the singer who insists on performing last. So there is only 1 way to arrange this performer. Thus, the total number of ways to arrange all performers is \(24*1 = 24\) ways.

Key Concepts

Factorial

The term 'factorial' is used in mathematics to describe the product of an integer and all the integers below it, down to one. For example, the factorial of 4 is denoted as 4! and calculated as:
\(4! = 4 \times 3 \times 2 \times 1 = 24\).
Factorials grow very rapidly with each added integer. They are central to calculating permutations because they represent the total number of ways in which a set of objects can be arranged in order.

Arrangements

An arrangement in mathematical terms refers to a sequence or ordering of objects where the order is important. In our case, the order of singers performing at the nightclub is significant. If even two singers swap their performance slots, it would be considered a different arrangement. The number of possible arrangements increases factorial with respect to the number of objects to arrange. It's important to note that when certain constraints are in place, like a singer insisting on being last, this affects the total number of arrangements, as some possibilities are eliminated.

Permutation Formula

The permutation formula is used to find the number of possible arrangements when the order of those arrangements matters. The formula for the permutation of n distinct objects is n!. This translates to multiplying the number of available options (n) by the factorial of one less than the number of options (n-1), and so on, down to one.
In our nightclub singer example, even though there are 5 singers, one insists on going last, reducing the problem to calculating the arrangements of the remaining 4 singers. The permutation formula simplifies to 4!, which is the factorial of 4, leading us to 24 possible arrangements before fixing the last performer's position.