Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ P(5,5) $$

Step-by-Step Solution

Answer

The number of permutations of a set of size 5, taken 5 at a time, is 120. To calculate this, we used the formula \(P(5,5) = \frac{5!}{(5-5)!}\), resulting in \(P(5,5) = \frac{120}{1}\), which simplifies to \(P(5,5) = 120\).

Step 1: Identify the values of m and n

In this case, we have m = 5 and n = 5. We will use these values in the formula for permutations to evaluate P(5,5).

Step 2: Apply the permutation formula

Using the values of m and n, we can plug them into the permutation formula: $$ P(5,5) = \frac{5!}{(5-5)!} $$

Step 3: Compute the factorials

Let's compute the factorials for both the numerator (5!) and the denominator (0!): $$ 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 $$ Note that 0! is defined to be equal to 1. Therefore, our expression becomes: $$ P(5,5) = \frac{120}{1} $$

Step 4: Calculate the final result

Finally, we just need to divide the numerator by the denominator to get the final result: $$ P(5,5) = \frac{120}{1} = 120 $$ The number of permutations of a set of size 5, taken 5 at a time, is 120.

Key Concepts

Factorial Notation

In understanding permutations, it's essential to start with the concept of factorial notation, symbolized by the exclamation point (!). A factorial represents the product of all positive integers from a given number down to 1. For example, the factorial of 5, denoted as \(5!\), is calculated by multiplying all whole numbers descending from 5 to 1:
\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

An important convention is that \(0!\) is always equal to 1, based on the definition of the factorial operation. This might seem counterintuitive, but it ensures consistency across various mathematical formulas, especially in combinatorics and permutations.

The factorial concept is central to calculating permutations because it indicates the total number of ways to arrange a set number of items without repetition. Understanding factorials provides a foundational block to further grasp the permutation formula.

Permutation Formula

Building upon factorial notation, we arrive at the permutation formula, a key tool for solving combinatorial problems involving the arrangement of items. The permutation formula is written as \(P(n, k)\), where \(n\) is the total number of items and \(k\) is the number of items to be arranged.

The formula is as follows:
\[P(n, k) = \frac{n!}{(n-k)!}\]
This equation calculates the number of distinct ways \(k\) items can be selected and arranged from a larger set of \(n\) items. In our exercise, \(P(5,5)\), we are looking at the total number of ways to arrange 5 items out of 5, which intuitively is a straightforward arrangement of those 5 items in every possible order - leading us to simply calculate \(5!\).

By plugging in the values for \(n\) and \(k\) into the formula, we can solve for the permutations. This formula is potent for evaluating expressions where the number of arrangements is vast and cannot be counted through simple enumeration.

Combinatorics in Mathematics

Combinatorics is the branch of mathematics that deals with counting, arranging, and finding patterns within sets of elements. Permutations are a part of combinatorics that focuses specifically on the arrangement of these elements.

Permutations consider the order of arrangement to be important – for instance, the sequence of letters ABC is different from ACB. This distinction is crucial when solving problems related to permutations. Combinatorics encompasses other topics like combinations, where the order does not matter, and various principles that help solve more complex problems.

By mastering factorial notation and the permutation formula, students can tackle a wide range of problems within combinatorics, from simple arrangements to more intricate structures like graphs and networks. This field of study is not only fascinating but also has practical applications in areas such as computer science, statistics, and operations research. Understanding the basics of combinatorics can be beneficial for students as it enhances their analytical and problem-solving skills, providing tools that are useful in numerous real-world scenarios.