Problem 75 - Basic Statistics for Business and Economics with Student

Question

.A computer password consists of four characters. The characters can be one of the 26 letters of the alphabet. Each character may be used more than once. How many different passwords are possible?

Step-by-Step Solution

Answer

There are 456,976 different possible passwords.

Step 1: Understand the Problem

We need to find how many combinations are possible for a password consisting of 4 characters, where each character can be one of the 26 letters in the alphabet and repetition of characters is allowed.

Step 2: Determine Choices per Character

Each character in the password has 26 possible choices because there are 26 letters in the alphabet. Since repetition is allowed, these choices are independent for each character.

Step 3: Calculate Total Combinations

The total number of possible passwords is calculated by multiplying the number of choices for each character. So, we have:\[ 26 \times 26 \times 26 \times 26 \] or equivalently, this can be written using exponents:\[ 26^4 \]

Step 4: Perform the Calculation

Calculate \( 26^4 \):\[ 26^4 = 26 \times 26 \times 26 \times 26 = 456976 \]

Step 5: Conclude with the Total Number

There are 456,976 different possible passwords that can be formed under these conditions, where each character can be repeated and there are 4 characters in the password.

Key Concepts

Password Combinations

Creating secure passwords is crucial, and understanding how many different passwords can be formed is a foundational concept in combinatorics. In this specific problem, we look at forming a password made up of four characters. Each character is chosen from the English alphabet, which consists of 26 letters: from A to Z. Given that each character can be repeated, the choices for each position in the password are independent. This independence means that any letter qualified to fill one position is equally qualified for another.

So, for each of the four positions in the password, there are 26 possibilities. The freedom to repeat characters significantly increases the total number of possible combinations, enhancing password strength against guessers. With repetition, exploring all possible combinations becomes manageable through simple multiplication, leading us to a staggering number of potential passwords.

Repetition in Permutations

Repetition plays a key role in permutations, especially in scenarios like password creation. In combinatorics, if characters or items in a sequence can be repeated, the total permutations multiply because each position can hold any of the allowable options independently, without regard to what fills the other positions.

This contrasts with situations where repetition isn't allowed, significantly reducing possible combinations as more positions are filled. In our password example, since each character can appear more than once, repetition results in more password variations. Plugging into the formula, each of the four positions drawing from the 26-letter set is handled as separate events.

• If repetition was not allowed, after choosing one letter, it could not be reused, reducing options for the next character.
• Allowing repetition keeps the options steady at 26 for each place, making calculations straightforward and opening a vast array of combinations.

Thus, the effects of repetition in permutations vastly increase possibilities when constructing passwords or any similar task with repeated elements.

Exponential Notation in Counting

Exponential notation is a powerful mathematical tool that simplifies counting, especially when dealing with large numbers of combinations. In our password combination scenario, instead of writing out the multiplication 26 times for four character slots, exponential notation provides a succinct representation.

The expression \(26^4\) efficiently captures this situation. Why use exponents here? Because exponents compactly express repeated multiplication.

• The base of the exponent (26) represents the number of choices available for each character.
• The power (4) indicates how many times these choices are made independently, corresponding to the four positions in our password.

This approach not only simplifies writing and calculation but also illustrates password strength growth exponentially. With exponential growth, even modest increases in length or options dramatically amplify possibilities, underlying the importance of length in strong password formulation.