Problem 63 - Algebra and Trigonometry

Question

How many different four-letter passwords can be formed from the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F},\) and \(\mathrm{G}\) if no repetition of letters is allowed?

Step-by-Step Solution

Answer

There are 840 different four-letter passwords that can be formed from the letters A-G without repetition.

Step 1: Identify the Parameters

Here, the total number of elements, n, is 7 (the letters A-G), and the number of elements to choose, k, is 4 (the length of the password). Remember that because this is a permutation problem, order matters.

Step 2: Apply the Permutations Formula

The permutation formula is \(nPr = n! /(n - r)!\). Apply this formula with n=7 and r=4.

Step 3: Calculate the number of Permutations

Substitute the given values into the formula. The calculation will be \(7P4 = 7! / (7 - 4)!\). Here, '!' stands for factorial, which is the product of all positive integers up to that number. So, 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 and 3! = 3 * 2 * 1.

Step 4: Final Calculation

Perform the calculation: 7! / 3! = (7 * 6 * 5 * 4 * 3 * 2 * 1) / (3 * 2 * 1) = 7 * 6 * 5 * 4 = 840.

Key Concepts

Factorial

Factorials are an essential concept in permutations and combinatorics. When you see a number followed by an exclamation mark, like 7!, this is the symbol for factorial. It represents the product of that number and all positive integers below it. For example, 7! means you multiply 7 by every integer below it, namely 6, 5, 4, 3, 2, and finally 1, making the computation:
- 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1.
This concept helps in calculating permutations and combinations, making it easier to understand complex arrangements of objects.
In the given problem, knowing how to calculate factorials allows us to find how many ways we can arrange the 4-letter passwords.

Combinatorics

Combinatorics is the field of mathematics dealing with combinations and arrangements of objects. It is fundamental in solving problems related to permutations and combinations. Combinatorics helps us calculate the number of ways to select items from a larger pool of objects, considering their arrangement and order.
In the context of the exercise, we are using combinatorics to determine all the possible four-letter passwords we can create from the letters A to G.

• Permutations, a topic within combinatorics, are utilized here because the order of letters matters when forming passwords.
• Using the principle of permutations, we consider different sequences for selecting our letters to form distinctive passwords.

Order of Selection

In permutation problems like this, the order in which items are selected is very important. Unlike combinations, where order doesn't matter, permutations take into account all possible orderings of selections.
Let's imagine selecting 4 letters to make a password. The password "ABCD" is entirely different from "ACBD" due to their sequences.
Thus, when solving permutation problems, the order of selection is vital to accurately calculating the total number of possible arrangements. The formula for permutations reflects this, ensuring all possible sequences are covered.

No Repetition Allowed

The constraint 'no repetition allowed' means that each letter can only be used once in creating a password. This rule alters the way we calculate permutations because it limits choices for each subsequent selection:

• In the first position, you can choose from any of the 7 letters.
• After one letter is used, only 6 letters remain for the next choice.
• This pattern continues until all positions are filled.

This restriction simplifies the permutation calculations, as we multiply the number of available choices for each step. Without repeating letters, each arrangement is unique, which streamlines and reduces the total combinations possible.