Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the letters \(\mathbf{A}, \mathbf{B}, \mathbf{C},\) and \(\mathbf{D}\). Write all permutations of the letters.

Step-by-Step Solution

Answer

The 24 permutations of the letters A, B, C, and D are: ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA, BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA.

Step 1: Understanding permutations

Permutation refers to the arrangement of objects without repetition where order is important. The number of permutations of \(n\) distinct objects is given by \(n!\), where '!' denotes factorial. For this problem, the four letters \( \mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D} \) are distinct, so there are \(4! = 4 \times 3 \times 2 \times 1 = 24\) permutations.

Step 2: Listing permutations

The 24 permutations of the letters \( \mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D} \) are: ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA, BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA. The key here is to start with one letter and permute the remaining ones, then move on to the next letter, and so on.

Key Concepts

Factorial Notation

Factorial notation is a mathematical way to express the product of all positive integers up to a particular number. It is symbolized by an exclamation mark placed after the number. For example, when we see \(4!\), we read it as "four factorial." This means multiplying 4 by every whole number less than itself down to 1, which gives us:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Factorials are crucial in permutations calculations because they help us figure out the number of ways objects can be arranged. When dealing with permutations of distinct objects, factorial notation provides a quick method to determine the number of possible arrangements.

Distinct Objects

In permutations, the concept of distinct objects implies that each object is unique and distinguishable from the others. This uniqueness means each object holds a particular place in any arrangement.

• For instance, with the letters \( \mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D} \), since all are distinct, different permutations or orderings become possible.

No two objects can be indistinct in this context, as that would alter the number of possible permutations. Imagine having two indistinct objects; this would reduce the total permutations since swapping places doesn't result in a new arrangement.

Order in Permutations

Order plays a vital role in permutations and differentiates it from combinations, where order doesn't matter. When we talk about permutations, we’re focusing on the different ways to arrange distinct objects.

• For example, "ABCD" is different from "ACBD" because the sequence in which the letters appear changes the permutation.

The step-by-step solution shows how starting with one letter and rearranging the remaining ones creates unique permutations, totaling 24 for the set \( \mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D} \). Each rearrangement respects the order, so modifying even one letter position leads to a new permutation.