Problem 2 - Algebra for College Students

Question

How many different nine-letter arrangements can be formed from the nine letters of the word APPARATUS? \(\quad 30,240\)

Step-by-Step Solution

Answer

45,360 distinct arrangements can be formed.

Step 1: Identify the Problem

We want to find the number of distinct nine-letter arrangements that can be made using the letters from the word 'APPARATUS'. This is a permutation problem involving repeating elements.

Step 2: Determine the Total Number of Letters

The word 'APPARATUS' consists of 9 letters, which we need to arrange. These include repeated letters which are: two 'A's, two 'P's, and two 'T's.

Step 3: Calculate the Total Permutations Without Repetition

First, calculate the total number of arrangements if all letters were different. This would be the factorial of the number of letters: \[ 9! = 362,880 \]

Step 4: Adjust for Repeated Letters

Divide the total permutations by the factorial of the frequency of each repeating letter to account for those repetitions:1. The letters 'A', 'P', and 'T' each repeat twice.So, divide by \[(2!)^3 = 2! \times 2! \times 2! = 2 \times 2 \times 2 = 8\]Thus, the number of distinct permutations is:\[ \frac{9!}{(2!)^3} = \frac{362,880}{8} = 45,360 \]

Key Concepts

Factorials and Their Importance

When you're tasked with arranging different objects, like letters, numbers, or any other elements, the concept of "factorials" becomes essential. Factorials help us determine how many different ways we can arrange a set of items. The notation for a factorial looks like this: \[ n! \] Where \(n\) is a whole number, and the "!" denotes factorial. For example, if you have 3 different items, the factorial \(3!\) tells you the number of different arrangements. To compute 3!, you multiply 3 by all the whole numbers below it:\[ 3! = 3 \times 2 \times 1 = 6 \]This means there are 6 possible ways to arrange 3 items.
A key factor here is understanding that as the number of items increases, the number of possible arrangements grows very quickly due to the multiplication of descending integers.

• Factorials provide the "total permutations" when each item in a set is distinct.
• They form the backbone of more complex arrangements, especially when dealing with repetitions, as in the exercise above.

The Magic of Combinatorics

Combinatorics is the branch of mathematics focused on counting, arrangement, and combination of sets. Understanding combinatorics is pivotal in problems where permutations with repetition arise.
When we talk about permutations, we essentially mean the arrangement of a set in a particular order. The role of combinatorics is to help us calculate these arrangements more effectively by consideration of various factors, like repetitions.

• Permutations are an arrangement of all elements in a particular order. When elements repeat, we adjust our calculations accordingly.
• Without the language of combinatorics, solving these problems systematically would be much more difficult and less intuitive.

By using combinatorics, you're able to take into account each element's occurrence and adjust permutations appropriately, as seen in the original exercise. This allows us to derive accurate counts for numerous problems, especially those involving repeated elements.

Understanding Distinct Arrangements

When solving permutation problems, the biggest challenge is often creating distinct arrangements, especially with repeated items. In our exercise, the word "APPARATUS" contains repeating letters: two 'A's, two 'P's, and two 'T's.

• Without these repetitions, you would calculate arrangements through a simple factorial of all letters.
• However, with repetitions, the permutations must be adjusted to ensure that arrangements forming the same sequence with different repetitions are only counted once.

This adjustment is performed by dividing the total number of permutations by the factorial of the frequency of each repeating element:\[ \frac{9!}{(2!)^3} = \frac{362,880}{8} = 45,360 \]Here, we account each repeated pair by dividing the total permutation count, ensuring that arrangements of 'APPARATUS' are unique. This adeptly highlights the difference between total number of arrangements and distinct arrangements, especially in words or sequences with repeating items.