Problem 30 - Algebra for College Students

Question

How many different ten-letter permutations can be formed from five identical A's, four identical B's, and one C? 1260

Step-by-Step Solution

Answer

There are 1260 different ten-letter permutations.

Step 1: Understanding the Problem

We need to find the number of different permutations of a sequence consisting of 10 letters: five 'A's, four 'B's, and one 'C'. Since some of the letters are repeated, this affects the total count of unique permutations.

Step 2: Applying the Permutation Formula for Repeated Elements

For permutations of a multiset, we use the formula:\[\text{Total permutations} = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}\]where \(n\) is the total number of items to arrange, and \(n_1, n_2, ..., n_k\) are the frequencies of the identical items. Here, \(n = 10\) (total letters), \(n_1 = 5\) (for 'A's), \(n_2 = 4\) (for 'B's), \(n_3 = 1\) (for 'C').

Step 3: Calculating the Factorials

Calculate the factorials needed:- \(10! = 3628800\)- \(5! = 120\)- \(4! = 24\)- \(1! = 1\)

Step 4: Plug Values into the Formula

Substitute the factorial values into the permutation formula:\[\frac{10!}{5! \, \cdot \, 4! \, \cdot \, 1!} = \frac{3628800}{120 \, \cdot \, 24}\]

Step 5: Compute the Number of Permutations

First, multiply the factorials of the repeated elements: \(120 \times 24 = 2880\).Then divide the total factorial by this product:\[\frac{3628800}{2880} = 1260\]This is the total number of different permutations possible.

Key Concepts

Understanding Factorials

In mathematics, a factorial is a way to express the product of a series of descending natural numbers. Imagine it like diving into a countdown where you multiply every number you pass by until you reach 1. The notation is an exclamation mark following a number, like this: \( n! \). So, for 5, the factorial \(5!\) is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials are incredibly useful in different mathematical scenarios, especially in permutations and combinations. They help us determine the number of ways to arrange items, which is fundamental to calculating permutations.
Remember these key points:

• \(0! = 1\), a definition that helps in various calculations.
• Factorials grow fast; for example, \(10!\) is already 3,628,800.
• Each step represents the multiplying by one less number, never going below 1.

Factorials give structure to the complex world of permutations and combinations, aiding in organizing objects systematically.

Multiset Permutations

Multiset permutations are a fascinating aspect of combinatorial mathematics. They address scenarios where we need to arrange elements that contain repetitions. Unlike simple permutations where each item is distinct, multisets consider identical items, requiring a different approach.
To find the number of permutations for a multiset, we use a specific formula:
\[\text{Total permutations} = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}\]
This formula helps account for the repetition of items. Here, \(n\) represents the total number of items we have, while \(n_1, n_2, ..., n_k\) correspond to the counts of each repeated item. Using this approach simplifies calculations in cases like the exercise, where letters repeat frequently.
Consider:

• Five identical 'A's.
• Four identical 'B's.
• One unique 'C'.

This arrangement affects how we count possible sequences since swapping identical elements doesn't yield a new permutation. Multiset permutations ensure we count only unique sequences.

Introduction to Combinatorial Mathematics

Combinatorial mathematics involves the study of counting, arranging, and combination of objects. It's a fundamental field for understanding diverse mathematical problems tied to arrangement and logic.
Combinatorics looks at how to gather objects within a set in particular ways, answering questions about the most efficient arrangements possible.
Key areas in this field include:

• **Permutations**, which are ways to arrange objects in order. They're crucial for understanding sequence and structure.
• **Combinations**, focusing on the selection of items without regard for their order.
• **Graph theory**, involving points connected by lines, dealing with networks and paths.

Each concept provides solutions to different types of problems, ranging from simple order arrangement to complex network systems. Think of combinatorial mathematics as a toolkit for solving real-world problems. Whether designing a network or understanding probability, combinatorics offers the solutions needed to logically approach and solve these problems.