Problem 30 - College Algebra

Question

Use the Fundamental Counting Principle to solve Exercises \(21-32\). 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 singers' performances.

Step 1: Arrange the Remaining Singers

Start by arranging the remaining four singers, those who are other than the one who insists on performing last. Since the order in which these singers perform matters, we need to find the number of permutations. In general, the number of ways to arrange \(n\) distinct objects is \(n!\) (n factorial), thus, there are \(4!=4*3*2*1=24\) ways to arrange the four other singers.

Step 2: Fix the Last Singer's Position

Next, we take account of the singer who insists on performing last. As there is only one way to place this singer in the last spot, we multiply the number of ways to arrange the first four singers by one.

Step 3: Apply the Fundamental Counting Principle

Finally, apply the Fundamental Counting Principle, which tells us to multiply together the number of ways to do each independent task. Here, we multiply the \(24\) ways to arrange the first four singers by the \(1\) way to position the last singer. As such, there are \(24*1=24\) different ways to schedule the appearances of the singers.

Key Concepts

Permutations

When we talk about permutations, we are discussing the different ways in which a set of objects can be ordered or arranged. Imagine you have a set of items and you want to know how many different sequences or orders you can create with them.
For example, with five distinct singers, the total permutations, or different arrangements, we could make without restrictions would be calculated as the factorial of 5, which is denoted as 5!. But in our exercise, we have a constraint: one singer must always be last in line.
To solve such problems, we often need to focus only on a subset of the objects, which in this case means the four other singers. Thus, we find out how many different permutations can be made with these four singers alone.

Factorial Notation

Factorial notation is a mathematical concept used to simplify the description of permutations. Written as an exclamation mark (!) following a number, factorial notation represents the product of all positive integers up to that number.
For instance, 4! is shorthand for the multiplication 4 × 3 × 2 × 1, which equals 24. This tells us there are 24 unique ways to organize four distinct items.
Using factorial notation is particularly crucial in combination with the Fundamental Counting Principle, especially in scenarios involving permutations of groups with various constraints or sequences.

Arrangement

The concept of arrangement focuses on how the order of items is important. Every time you alter the order of a set of items, even if the items themselves haven't changed, the arrangement is different.
In the case of the exercise, the arrangement has a specific constraint: one of the singers must always perform last. Hence, we arrange the other four singers first in all possible orders, which we calculated to be 24 ways, thanks to permutations and factorial notation.
Finally, we place the insistent singer in the last spot, confirming this arrangement maintains the pre-stated constraint of the exercise while demonstrating the effective use of the Fundamental Counting Principle.