Problem 32 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

25

Step 1: Identify the Total Number of Letters

There are 5 letters available: A, B, C, D, and E.

Step 2: Determine the Number of Positions in the Code

The code consists of 2 positions for letters.

Step 3: Calculate the Number of Choices for Each Position

For each of the 2 positions in the code, there are 5 possible letter choices.

Step 4: Apply the Counting Principle

Using the counting principle, multiply the number of choices for the first position by the number of choices for the second position: \[ 5 \text{ choices for the first position} \times 5 \text{ choices for the second position} = 25 \text{ possible codes} \]

Key Concepts

Counting Principle

The Counting Principle is a fundamental concept in combinatorics. It helps us determine the total number of possible outcomes in a multi-step process. In essence, if there are multiple stages to form an entity, and if each stage has a set number of options, simply multiply the number of options at each stage to find the total number of possible entities.
For example, if you want to create a two-letter code with 5 different letters, you can use this principle:

• There are 5 choices for the first letter.
• There are also 5 choices for the second letter.

Therefore, you multiply 5 (choices for the first letter) by 5 (choices for the second letter) to get 25 possible codes. This logic applies to any number of stages and options.

Permutation

Permutation refers to an arrangement of all members of a set into some sequence or order. When dealing with permutations, order matters.
However, in the context of forming codes, we need to consider how we're choosing and arranging elements. In this exercise, we're choosing 2 letters from a set of 5 (A, B, C, D, E). Since the order in which we place the letters matters (AB is different from BA), we are dealing with permutations.
To compute the number of possible permutations when there is a specific number of positions (like our two-letter code), we use the Counting Principle. Each letter can appear in each position, yielding the total permutations as the product of possible choices at each step. Thus, the number of permutations for our scenario, considering repetitions, is 25, as calculated with the Counting Principle.

Repetition Allowed

In many combinatorial problems, the concept of whether repetition is allowed changes the complexity and the approach. When repetition is allowed, the same element can be chosen more than once for different positions.
For our two-letter code problem, repetition is allowed. This means that a code like 'AA' or 'BB' is perfectly valid.

• With repetition, each choice is independent of the previous choices.
• Every position in your code can be any of the 5 letters (A, B, C, D, E).

Without repetition, the calculation would be different because we would then subtract available choices after the first selection. But since repeating is permitted, we maintain the same number of choices for each position, hence 5 choices per position.
This leads us back to multiplying the number of choices by itself for each position, so 5 choices per position multiplied by 2 positions results in 25 combinations.