Question

Rosita was trying to find a relationship between the number of letters in a word and the number of different ways the letters can be arranged. She considered only words in which all the letters are different. $$\begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Number of } \\ \text { Letters } \end{array} & \text { Example } & \begin{array}{c} \text { Number of } \\ \text { Arrangements } \end{array} \\ \hline 1 & \mathrm{A} & 1(\mathrm{A}) \\ 2 & \mathrm{OF} & 2(\mathrm{OF}, \mathrm{FO}) \\ 3 & \mathrm{CAT} & 6(\mathrm{CAT}, \mathrm{CTA}, \mathrm{ACT}, \mathrm{ATC}, \mathrm{TAC}, \mathrm{TCA}) \\ \hline \end{array}$$ a. Continue Rosita's table, finding the number of arrangements of four different letters. (You could use MATH as your example, since it has four different letters.) b. Challenge Predict the number of arrangements of five different letters. Explain how you found your answer.

Short answer

For four letters, there are 24 arrangements; for five letters, there are 120.

Step-by-step solution

Title - Analyzing the existing pattern

Look at the given examples in the table to understand the pattern. For 1 letter, there is 1 arrangement. For 2 letters, there are 2 arrangements. For 3 letters, there are 6 arrangements.

Title - Finding the number of arrangements for four letters

Use the factorial function to determine the number of arrangements. For four different letters (e.g., MATH): The formula for the number of arrangements is given by the factorial of the number of letters. Thus, for 4 letters: \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Title - Understanding the factorial concept

Recognize that the pattern follows the factorial function. The factorial of a number n, denoted as \(n!\), is the product of all positive integers less than or equal to n. For example, \(3! = 3 \times 2 \times 1 = 6\). This confirms the number of arrangements for 3 letters.

Title - Predicting the number of arrangements for five letters

Using the factorial concept again, determine the number of arrangements for five different letters. For five letters, such as in the word SNAIL: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). So, there would be 120 different arrangements.

Key concepts

Factorials

When you look at different word arrangements, you'll see a pattern that follows what's called a 'factorial'. A factorial is a mathematical operation where you multiply a series of descending natural numbers.
To write a factorial for a number, you put an exclamation mark after it. For example, the factorial of 4 is written as 4!.
To find the value of 4!, you multiply all whole numbers from 4 down to 1:
\[4! = 4 \times 3 \times 2 \times 1 = 24 \]
This applies to any number, so if you had 6 letters, it would be 6!:
\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\]
Factorials are extremely helpful in counting arrangements and understanding permutations.

Arrangements

Arrangements, or permutations, are different ways you can order a set of items. If you have the letters A, B, and C, there are several ways to arrange them: ABC, ACB, BAC, BCA, CAB, and CBA.
The total number of different arrangements can be found using the factorial function.
If you have 3 letters, you'd find it by calculating 3!:
\[3! = 3 \times 2 \times 1 = 6\]
This tells us there are 6 unique ways to arrange the letters.
If you want to arrange 4 letters, like in the word MATH, it's slightly more complex, but the concept is the same:
\[4! = 4 \times 3 \times 2 \times 1 = 24\]
So there are 24 different ways to arrange the letters in MATH.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects. It's very useful in solving problems involving arrangements.
When you're arranging different items, or 'permutations', you're often using combinatorics.
For instance, with 5 letters like in the word SNAIL, combinatorics helps us find the number of ways to arrange the letters:
We use the factorial of 5 (5!), which is calculated as:
\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
This gives us 120 unique arrangements. Whether arranging letters, scheduling events, or dealing with passwords, combinatorics provides the tools to count all possibilities.

Mathematical Patterns

Mathematical patterns help us predict outcomes and solve problems more efficiently. Understanding the pattern between the number of letters and their arrangements using factorials is one such example.
Let's recap the pattern discovered in the exercise:
- 1 letter: 1 arrangement
- 2 letters: 2 arrangements
- 3 letters: 6 arrangements
- 4 letters: 24 arrangements
You can see that each step follows the factorial function.
These patterns give us a shortcut to find answers without listing every possibility.
By recognizing that n letters can be arranged in n! ways, we can handle larger problems with ease.
For example, with 6 letters, follow the same factorial pattern:
\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\]
So, always look for these kinds of patterns in mathematics—they make complex tasks simpler!