Problem 23 - Algebra 2 and Trigonometry

Question

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. FACTOR

Step-by-Step Solution

Answer

There are 720 different arrangements for the letters in the word "FACTOR."

Step 1: Identify the number of letters

First, we need to determine how many letters are in the word "FACTOR." There are 6 letters: F, A, C, T, O, R.

Step 2: Check for repeated letters

Next, we check if any letters are repeated in the word "FACTOR." Since all the letters are distinct, there are no repeated letters to account for.

Step 3: Calculate the total arrangements

Since there are 6 distinct letters, the total number of different arrangements is found by calculating the factorial of 6. The formula for this is: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Step 4: Simplify the factorial expression

Now, let's perform the multiplication to solve the factorial expression: \[ 6 \times 5 = 30 \]\[ 30 \times 4 = 120 \]\[ 120 \times 3 = 360 \]\[ 360 \times 2 = 720 \]\[ 720 \times 1 = 720 \]

Step 5: State the final result

The total number of different arrangements for the letters in the word "FACTOR" is 720.

Key Concepts

Understanding Factorials

Factorials are a key concept in calculating permutations and arrangements. A factorial is represented by the symbol "!" and indicates the product of all positive integers up to that number. For example, the factorial of 6 is written as \(6!\). To calculate \(6!\), multiply each whole number less than or equal to 6: \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\] This operation results in 720, which means there are 720 different ways to arrange 6 items.

Factorials are used in permutations when dealing with distinct items because they help account for every possible order of these items. This is crucial for solving problems like arranging letters in a word.

What Makes Letters Distinct?

In the context of permutations, distinct letters mean that each letter is unique in the arrangement. No letter appears more than once.

In our example with the word "FACTOR," each letter is different, meaning all of them contribute individually to the total number of arrangements without needing to account for repeats. This simplicity makes it straightforward to calculate permutations using the factorial of the total number of letters. If a word had repeated letters, we'd need to adjust our calculations, dividing the factorial by the factorial of the number of repeated letters to account for indistinguishable arrangements.

Arrangements and Their Calculations

Arrangements refer to the different ways we can organize a set of items. When all items are distinct, like the letters in "FACTOR," calculating the number of arrangements becomes straightforward by using permutations.

Typically, you start by determining the total number of items (letters, in this case). Then, since each letter is treated as unique, you can directly calculate the number of permutations using factorials. The result gives you the total number of possible orderings.

Here's a step-by-step for arranging distinct items:

• Count the total number of distinct items.
• Apply the factorial to this total: \(6!\).
• Solve the factorial equation for the total permutations.

These steps are essential for problems where every item is different, ensuring that you capture every possible sequence without missing or duplicating any scenario.