Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

How many four-letter permutations can be formed from the first four letters of the alphabet?

Step-by-Step Solution

Answer

There are 24 different four-letter permutations that can be formed from the first four letters of the alphabet.

Step 1: Understand the permutation formula

The formula for permutation is denoted as P(n, r), which represents the number of possible permutations when we choose 'r' items from a set of 'n' items where the order of the items matters. The formula is given by P(n, r) = n! / (n - r)!, where 'n!' denotes the factorial of the number 'n' and n! = n * (n-1) * (n-2) * ... * 1.

Step 2: Identify n and r

In this exercise, we have 4 letters (A, B, C, and D) to choose from, so n = 4. We want to form a four-letter permutation, which means we are selecting all the 4 letters; therefore, r = 4.

Step 3: Apply the permutation formula

Now substitute the values of n and r into the permutation formula: P(n, r) = n! / (n - r)! = P(4, 4) = 4! / (4 - 4)!.

Step 4: Calculate the factorials

Find the factorials of the numbers involved in the formula: \[\] 4! = 4 * 3 * 2 * 1 = 24 \[\] 0! = 1 (As per the definition of factorial, 0! = 1) \[\] Now substitute these values back into the formula: P(4, 4) = 24 / 1.

Step 5: Calculate the final result

Now calculate the final result for the formula: P(4, 4) = 24/1 = 24. Thus, there are 24 different four-letter permutations that can be formed from the first four letters of the alphabet.

Key Concepts

Permutation Formula

The permutation formula is a mathematical expression used to calculate the number of ways a subset of items can be arranged where the order is important. It's a cornerstone concept in the study of combinatorics, which provides us with the tools to count, arrange, and reorganize items within a set in numerous ways.

When we dive into the specifics, the permutation formula is denoted as P(n, r), where 'n' represents the total number of items to choose from, and 'r' stands for how many items we want to select from that set. The formula is expressed as \( P(n, r) = \frac{n!}{(n - r)!} \). It's essential to note that in permutations, changing the order of the selected items gives us a new permutation, signifying that order is crucial.

In the provided exercise, we're working with the first four letters of the alphabet, where all four letters are selected, indicating both n and r equal to 4. This direct scenario lets us utilize the formula, by simply calculating P(4, 4) to find that there are 24 unique ways to arrange these letters.

Factorial

The factorial, denoted by an exclamation point (!), is a fundamental concept in mathematics, predominantly in the field of combinatorics. A factorial represents the product of all positive integers up to a certain number. The mathematical definition of factorial for a non-negative integer n is \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \), with one peculiar instance where \(0! = 1\).

The factorial is pivotal when we calculate permutations because it tells us how many ways we can arrange 'n' objects in a total order. In the context of our exercise, while calculating P(4, 4), the factorial of 4 (4!) equates to 24. Factoring in the special case that 0! equals 1, we understand why the denominator of our permutation formula becomes 1, leading us to our final answer without further complications.

Combinatorics

Combinatorics is an extensive field in mathematics focused on counting, arranging, and finding patterns in the combinations of distinct items. Its applications span simple counting to the analysis of complex and abstract algebraic structures. One of the primary reasons we study combinatorics is to understand and formulate the principles behind permutations and combinations.

Permutation, the arrangement of objects where order matters, is just one aspect of combinatorics. The subject also includes combinations, where order does not matter, and other complex structures such as graphs and networks, which have significant applications in computer science, statistics, and operations research.

Within combinatorics, we encounter problems like the one in our exercise, where we apply permutation formulas to solve real-world problems, providing a clear demonstration of the practical uses of combinatorial principles. This interplay of concepts underlies the rich and varied nature of combinatorics.