Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Classify each problem according to whether it involves a permutation or a combination. In how many ways can a six-letter security password be formed from letters of the alphabet if no letter is repeated?

Step-by-Step Solution

Answer

The problem is a permutation since the order of the letters in the password matters. There are 26 letters in the alphabet and we need to form a six-letter security password without repeating any letters. Using the formula \(P(n, k) = \frac{n!}{(n-k)!}\), we find that there are 3,315,312 ways to create this password.

Step 1: Identify problem type

This is a permutation problem since the order in which the letters appear in the password matters.

Step 2: Identify total number of letters

We have 26 letters in the alphabet.

Step 3: Decide the number of letters to use

We need to form a six-letter security password without repeating any letters.

Step 4: Calculate permutations

To calculate the number of permutations, we use the formula: \(P(n, k) = \frac{n!}{(n-k)!}\) where \(n\) is the total number of items, \(k\) is the number of items selected, and \(P(n, k)\) is the number of permutations. In our case, n = 26 (total number of letters in the alphabet) and k = 6 (number of letters in the password). Now, calculate the factorial values needed: \(26! = 26 \times 25 \times 24 \times ... \times 3 \times 2 \times 1\) and \((26-6)! = 20! = 20 \times 19 \times 18 \times ... \times 3 \times 2 \times 1\) Now, we can calculate the permutations: \(P(26, 6) = \frac{26!}{(26-6)!} = \frac{26!}{20!}\)

Step 5: Simplify and solve

Simplify by canceling out \(20!\) in both numerator and denominator: \(P(26, 6) = \frac{26 \times 25 \times 24 \times 23 \times 22 \times 21}{1}\) Now, multiply: \(P(26, 6) = 26 \times 25 \times 24 \times 23 \times 22 \times 21 = 3,315,312\)

Step 6: State the answer

There are 3,315,312 ways to create a six-letter security password from the alphabet without repeating any letters.

Key Concepts

Factorials

Factorials are fundamental in the world of permutations and combinations. A factorial, symbolized by an exclamation mark (!) after a number, represents the product of all positive integers up to and including that number.
For example, 5! is equal to 5 x 4 x 3 x 2 x 1 = 120.

Factorials are incredibly useful when dealing with problems that involve arranging or organizing different items. They help us calculate permutations and combinations.
Especially when no repetition is allowed, factorials come in handy for calculating the number of possible arrangements or selections of items.

Here is how factorial works in permutations:
* When calculating permutations of a set, it involves dividing a larger factorial by a smaller one to account for the ordered nature of these arrangements.
* For instance, as in the problem with a password of six letters from the alphabet, calculating 26! gives us the number of ways to arrange all 26 letters. However, we only care about a subset of these, so we divide by (26-6)! to find just the permutations of six unique letters.

Password Creation

Creating secure passwords is like creating permutations; the sequence in which characters appear matters. In our example, forming a six-letter password from 26 unique alphabet letters without repetition is precisely a permutation issue.

This is because each position in the password holds significance. A minor change in the order results in a completely different password, much like how altering the sequence of numbers changes a code or an ID.

For password security:
* Always remember the order matters; this is similar to arranging different blocks in a line. Each order creates a unique sequence, thus, maximizing security.
* Without repetition, once a letter is used, it is unavailable for another slot in the same password.
This increases the complexity and the number of permutations."

Combinatorial Analysis

Combinatorial analysis is a branch of mathematics that studies counting techniques and strategies for devising configurations of objects according to given criteria. It covers permutations, combinations, and related principles.
When selecting items with different constraints, it is important to understand if the arrangement (order) matters.
* Permutations are preferred when order matters, such as creating passwords where each letter occupies a unique position.
* Combinations are used when order does not matter; for example, selecting a group of items where the sequence is irrelevant.

In our password example, combinatorial analysis allows us to calculate permutations efficiently using factorial calculations to handle the sequential nature of the problem. Understanding whether you're dealing with a permutation or combination is a crucial step in solving these problems effectively.