Problem 31 - Algebra 2

Question

REVIEW Jessica wants to create several different 7 -character passwords. She wants to use arrangements of the first three letters of her name, followed by arrangements of 4 digits in 1987 , the year of her birth. How many different passwords can she create? $$ \begin{array}{llllllll}{F} & {672} & {G} & {288} & {H} & {576} & {} & {J} & {144}\end{array} $$

Step-by-Step Solution

Answer

The number of different passwords is 144.

Step 1: Understand the Problem

Jessica wants to create passwords using arrangements of the first three letters of her name "Jes" and the digits of the year 1987. We need to count the total possible arrangements.

Step 2: Permutation of Letters

Calculate the number of ways to arrange the letters "J", "E", and "S". Use the permutation formula for 3 distinct items: \(3! = 3 \times 2 \times 1 = 6\).

Step 3: Permutation of Digits

Calculate the number of ways to arrange the 4 digits from 1987, which are 1, 9, 8, and 7. Use the permutation formula for 4 distinct items: \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 4: Combine Arrangements

Multiply the number of arrangements of letters by the number of arrangements of digits to find the total number of passwords: \(6 \text{ (arrangements of letters)} \times 24 \text{ (arrangements of digits)} = 144\).

Key Concepts

Password Combinations

Creating password combinations can be both fun and challenging. Imagine Jessica trying to make a unique password using parts of her name and other numbers close to her heart. To get the total possible combinations, you simply multiply the number of ways to arrange different parts of the password.

For Jessica, the password splits into two parts: the letters "JES" from her name and the numbers "1987" from the year of her birth. Each part can have multiple arrangements. To figure out the total number of distinct passwords, you calculate the number of possibilities for each separate part, then multiply those numbers together. This ensures all possible variations are considered.

This arrangement of letters and numbers is a perfect example of a permutation problem, where the order in which you arrange the items matters. By understanding how to calculate permutations, you can create secure and unique passwords, just like Jessica wants.

Factorial Calculation

Factorials are a core concept in permutations. They help us calculate the total number of ways to arrange a set of distinct items. A factorial of any positive integer, represented by an exclamation mark (!), is the product of all positive integers up to that number.

For example, the factorial of 3 is calculated as:
\[ 3! = 3 \times 2 \times 1 = 6 \]
This means there are 6 different ways to arrange three distinct items. Similarly, the factorial of 4:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
gives us the number of ways to arrange four items.

Factorial calculations are crucial when solving permutation problems, like Jessica's need to arrange the letters and numbers for her password. Understanding how to calculate factorials allows you to solve similar problems effortlessly, giving you the foundation to handle more complex permutations.

Distinct Arrangements

Arranging distinct items involves understanding that each item is unique and its positioning within the set matters. In Jessica's scenario, the letters "J," "E," and "S" as well as the numbers 1, 9, 8, and 7 are all unique.

This uniqueness is the key to permutation problems. When items are distinct, every change in the order leads to a new arrangement, and thus a new password in Jessica's case.

Calculating distinct arrangements involves finding out how many ways you can order each separate group. For the letters, it was computed as 6 ways, and for the numbers, 24 ways. Combining these two distinct arrangements by multiplication gives all possible configurations, enabling Jessica to generate secure passwords.

Recognizing whether items are distinct or if repetition is allowed is fundamental to solving such problems efficiently. This knowledge not only aids in password creation but also extends to various applications involving permutations.