Problem 31 - Algebra and Trigonometry

Question

How many two-letter codes can be formed using the letters \(A, B, C,\) and \(D ?\) Repeated letters are allowed.

Step-by-Step Solution

Answer

There are 16 possible two-letter codes.

Step 1 - Understand the Problem

Identify that we need to form two-letter codes using the letters A, B, C, and D and that repeated letters are allowed.

Step 2 - Identify the Number of Choices for Each Position

Since there are 4 letters to choose from for the first position and 4 letters to choose from for the second position, each position in the code has 4 possible choices.

Step 3 - Calculate the Total Number of Codes

To find the total number of possible codes, multiply the number of choices for the first position by the number of choices for the second position. So, the calculation will be: \( 4 \times 4 = 16 \).

Key Concepts

two-letter codes

A two-letter code is a sequence that consists of exactly two characters. In this exercise, we are creating two-letter codes using the letters A, B, C, and D. Each code contains just two characters selected from these letters. The order of the characters is important in forming distinct codes. For example, the code 'AB' is different from 'BA'.
These kinds of codes are often used in different contexts, such as temporary passwords, labelings, or shorthand notations. By understanding how to form these, you can handle more complex combinatorial problems in the future.

repeated letters allowed

In this exercise, we are allowed to use the same letter more than once in a two-letter code. This means that combinations like 'AA', 'BB', etc., are valid. When repeated letters are allowed, it broadens the range of possible codes that can be formed.
This concept is often referred to as 'with replacement' in combinatorics, indicating that the pool of letters does not diminish after selecting the first character. Think of it as having an unlimited supply of each letter, ready to be picked again for the next position in the code.

multiplication principle

The multiplication principle is a fundamental rule in combinatorics used to determine the number of possible outcomes of a sequence of choices. The principle states that if you have multiple stages in a process, and each stage has a certain number of choices, the total number of outcomes is the product of the number of choices at each stage.
In our case, forming a two-letter code has two stages: selecting the first letter and selecting the second letter. Since we have 4 options (A, B, C, D) for the first letter and another 4 options for the second letter, we multiply these choices: \( 4 \times 4 = 16 \). This calculation illustrates how powerful and straightforward the multiplication principle is for solving combinatorial problems.