Problem 13 - Algebra for College Students

Question

How many permutations of the four letters A, B, C. and \(D\) can be formed by using all the letters in each permutation? 24

Step-by-Step Solution

Answer

There are 24 permutations of the letters A, B, C, and D.

Step 1: Understanding Permutations

Permutations refer to the different ways of arranging a set of items. For four distinct letters, each permutation is a unique sequence that uses all letters exactly once.

Step 2: Define the Number of Items

There are 4 distinct letters: A, B, C, and D. Thus, we have 4 items to arrange for each permutation.

Step 3: Calculate Factorial of 4

The number of permutations of 4 items is given by the factorial of 4, written as \(4!\). The factorial function means multiplying the sequence of descending integers from the defined number down to 1.

Step 4: Compute 4 factorial

Compute \(4! = 4 \times 3 \times 2 \times 1 = 24\). Thus, there are 24 different permutations when all letters are used in each permutation.

Key Concepts

Factorial

The concept of a factorial is pivotal in understanding permutations. A factorial, denoted by an exclamation mark \(!\), represents the product of all positive integers up to a certain number. For example, the factorial of 4, written as \(4!\), is the product \(4 \times 3 \times 2 \times 1\).

Factorials grow quickly, and they calculate the total number of ways you can order a certain set of items. Consider the number 0; its factorial is defined as 1. This might seem odd initially, but it makes sense in terms of mathematical definitions and functions. Understanding factorials helps in evaluating permutations, especially for arranging distinct items.

Arranging Items

Arranging items involves changing the order of objects or people in various sequences. In permutations, every item in a set must be included, and the order in which they appear matters.

For instance, given four letters like A, B, C, and D, each different order of these letters represents a unique permutation. Arrangements differ based on the sequence used, such as ABCD, BACD, or CADB.

Notably:

• All items must be used in each permutation.
• The sequence results in different outcomes.
• The position of the elements is crucial.

The result is influenced not just by which items you choose but how you choose to order them. This is why, with more items, we see exponentially more permutations.

Distinct Letters

When dealing with permutations, understanding 'distinct letters' is important. These are letters that are unique from one another, with no repeats in the set being arranged.

Having distinct letters like A, B, C, and D simplifies calculations because each one is different. This means each position in the permutation can be filled uniquely by these letters.

• No two letters are the same, eliminating potential recurring sequences.
• This distinct quality helps in straightforward calculation by using the factorial.
• Each new permutation is a unique combination not found elsewhere with the same letters.

In summary, dealing with distinct letters ensures that no overlap or repetition affects your permutation count, making computation clearer and results precise.