Problem 3 - Thinking Mathematically

Question

Does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem.) How many different four-letter passwords can be formed from the letters \(A, B, C, D, E, F\), and \(G\) if no repetition of letters is allowed?

Step-by-Step Solution

Answer

The number of unique four-letter passwords which can be formed from the letters A, B, C, D, E, F, G without repetition is 840.

Step 1: Understanding Permutations

Permutations refer to the arrangement of items where the order is important. In this case, we are dealing with seven different alphabets \(A, B, C, D, E, F, G\) and we need to form a four-letter password, where the order of these letters is important i.e., 'ABCD' is different from 'DCBA'

Step 2: Defining the problem in terms of Permutations

The problem can be defined as a permutation problem since we are selecting 4 alphabets out of the 7 available, to form a four-letter password, and the order of the alphabets is important. The general formula to calculate permutations is P(n, r) = n! / (n - r)! where n is the total number of items available, r is the number of items we are selecting, ! means factorial.

Step 3: Calculating the Permutations

In this case, we are given 7 different alphabets, so n = 7. We need a password of 4 letters, so r = 4. Substitute n and r into the formula, P(7, 4) = 7! / (7 - 4)! = 7! / 3! = 5040/6 = 840. Thus there are 840 different four-letter passwords that can be formed from the letters A, B, C, D, E, F, G without repetition.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and analyzing sets of items. It explores how items are selected and arranged, and is integral to solving problems related to permutations and combinations. Understanding the differences between these two is crucial when faced with problems involving arrangements like forming passwords. Combinatorics looks at two primary aspects:

• Permutations: Where order is important.
• Combinations: Where order is not important.

In the context of the given exercise, we use permutations because the sequence in which we arrange the letters matters – 'ABCD' is a very different password from 'DCBA'. So, combinatorics provides the tools to calculate the total number of possible passwords efficiently.

Factorial

The concept of factorial is central in combinatorics, especially for calculating permutations. Factorial, represented by an exclamation mark (!), denotes the product of all positive integers up to a particular number. For instance:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 3! = 3 × 2 × 1 = 6

Factorials simplify the computation of permutations. In the exercise, to form four-letter passwords from seven available letters, we use the formula for permutations: \[ P(n, r) = \frac{n!}{(n - r)!} \]This formula requires us to calculate the factorial of both the total number of items (n) and the difference between the total items and selected items (n-r). This allows us to determine the total permutations, in this case, producing 840 unique passwords.

Password Formation

When we talk about password formation, especially in combinatorics, we're discussing arrangements where sequence is key. For a password, even if the characters are the same, different orders mean different passwords. For example, 'ABCD' is distinct from 'DCBA'.

In the exercise, we had 7 possible letters and needed to create a combination of 4, without repeating any letters. The importance lies in how these letters are structured – that's why permutations, not combinations, are used. The formula \( P(7, 4) = \frac{7!}{3!} \)illustrates this process clearly. It accounts for the arrangement's significance and limits the selection to 4, resulting in 840 distinct outcomes. This is crucial in contexts like secure password creation, where variety and sequence directly influence security.

Finite Mathematics

Finite mathematics is the study of mathematical concepts that do not require the infiniteness of numbers, focusing on discrete values. Often found in business, computer science, and social sciences, it includes topics like logic, statistics, and combinatorics. Combinatorics is particularly vital as it deals with finite collections of objects, much like in the password problem.

In our example, finite mathematics helps us compute the possible arrangements of a set number of letters (7) into smaller groups (4), ensuring we stay within tangible, finite results. By using permutations, we navigate within these limits – effectively utilizing finite calculations like factorials to determine specific arrangements, a crucial part of crafting precise solutions in finite mathematics.