Problem 18 - Algebra 2

Question

Abby is registering at a Web site. She must select a password containing six numerals to be able to use the site. How many passwords are allowed if no digit may be used more than once?

Step-by-Step Solution

Answer

Abby can create 151,200 different passwords.

Step 1: Understanding the Problem

Abby needs to create a password using six unique digits. Each digit must be a numeral (0 through 9). We need to calculate how many different combinations of six numerals can be formed without repeating any digit.

Step 2: Determining the Total Numerals Available

There are ten numerals in total from 0 to 9. Abby can use these numerals to form her six-digit password.

Step 3: Choosing and Arranging the Numerals

We are choosing 6 numerals out of 10. Since order matters in a password, we need to calculate permutations of selecting and arranging these numerals. The formula for permutations is given by \( P(n, k) = \frac{n!}{(n-k)!} \), where \(n\) is the total number of items to choose from, and \(k\) is the number of items to choose.

Step 4: Calculating the Permutations

Substitute \(n = 10\) and \(k = 6\) into the permutations formula: \[ P(10, 6) = \frac{10!}{(10 - 6)!} = \frac{10!}{4!} \]. Simplifying, \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), and \(4! = 4 \times 3 \times 2 \times 1\), yielding \( \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5}{1} \).

Step 5: Simplifying the Calculation

Calculate the product: \(10 \times 9 = 90\), \(90 \times 8 = 720\), \(720 \times 7 = 5040\), \(5040 \times 6 = 30240\), and finally, \(30240 \times 5 = 151200\).

Step 6: Final Answer

The total number of different passwords Abby can create with six unique digits is 151200.

Key Concepts

Combinatorics

Combinatorics is an essential branch of mathematics focusing on counting, arranging, and combination of objects. Understanding this field helps solve problems related to arrangements and selections, similar to Abby's password problem. The key to combinatorics lies in categorizing problems into permutations and combinations.

• Permutations: This is concerned with the arrangement of objects where the order is important.
• Combinations: This is focused on selecting objects where the order doesn’t matter.

In Abby's case, since each digit must appear uniquely in her password, the calculation involves permutations. It's crucial to distinguish the nature of the task—arrangement or selection—to apply the correct formula. Combinatorics provides the tools we need to approach such problems systematically and confidently.

Factorial

Factorials are a fundamental component in permutations and combinations calculations.The factorial of a number, denoted by an exclamation mark (such as 5!), is the product of all positive integers up to that number.For example:

• 5!: Which means 5 × 4 × 3 × 2 × 1 = 120.
• 4!: Calculated as 4 × 3 × 2 × 1 = 24.

Factorials help compute the number of ways to arrange a set of items.In the case of Abby's password problem, we use factorials to determine permutations.Formula usage like \( P(n, k) = \frac{n!}{(n-k)!} \) utilizes factorials for both the numerator and denominator, helping us find different arrangements of chosen items.Being able to calculate factorials is crucial for solving many combinatorial problems efficiently.

Password Combinations

When creating password combinations, especially with constraints like uniqueness, permutations come into play due to the importance of order.Abby needs to construct a unique six-digit password using numerals 0-9, not repeating any character.The formula \( P(n, k) = \frac{n!}{(n-k)!} \) is perfect for these situations.For example, using this formula:

• Start with 10 digits: From 0 to 9.
• Select 6 digits: Order matters, no repetitions allowed.
• Compute using the permutations formula, \( P(10, 6) \).

Plugging in the values, the computation becomes \( \frac{10!}{4!} \) leading to 151,200 possible passwords.This large number of combinations ensures a secure and robust password.Understanding the permutations concept is pivotal in crafting passwords that are both secure and adherent to specific rules.