Problem 34 - College Algebra

Question

,*,#,@,@,\$,\$,\$,\%,\%,\%,\% that begin and end with "\%" # For the following exercises, find the distinct number of arrangements. The symbols in the string #,*,#,@,@,\$,\$,\$,\%,\%,\%,\% that begin and end with "\%"

Step-by-Step Solution

Answer

75,600 arrangements.

Step 1: Understand the Problem

We need to find the number of distinct arrangements for the string **#,*,#,@,@,$,$,$,%,%,%,%** that start and end with "%".

Step 2: Fix the Ending and Starting Symbols

Since the arrangement must begin and end with '%', the first and the last symbol is fixed, reducing the sequence to be arranged to **#,*,#,@,@,$,$,$,%,%**.

Step 3: Determine the Position of Symbols and Their Counts

From the remaining sequence **#,*,#,@,@,$,$,$,%,%**, we have: - 2 '#' symbols - 1 '*' symbol - 2 '@' symbols - 3 '$' symbols - 2 '%' symbols

Step 4: Calculate the Number of Arrangements

The total number of arrangements of the symbols **#,*,#,@,@,\(,\),\(,%,%** can be calculated using the formula for permutations of multiset:\[ \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!} \]In this case:\[ n = 10 \text{ (total symbols)} \]\[ n_1 = 2 \text{ (for '#')}, \, n_2 = 1 \text{ (for '*')}, \, n_3 = 2 \text{ (for '@')}, \, n_4 = 3 \text{ (for '\)')}, \, n_5 = 2 \text{ (for '%')} \]Substituting these into the formula:\[ \frac{10!}{2! \times 1! \times 2! \times 3! \times 2!} \]

Step 5: Compute the Factorials

Calculate each factorial:- **10! = 3,628,800**- **2! = 2**- **1! = 1**- **3! = 6**Now substitute back into the arrangement formula:\[ \frac{3,628,800}{2 \times 1 \times 2 \times 6 \times 2} = \frac{3,628,800}{48} \]

Step 6: Final Calculation

Perform the division from Step 5:\[ \frac{3,628,800}{48} = 75,600 \]Therefore, there are **75,600** distinct arrangements of the symbols that start and end with '%'.

Key Concepts

Factorials

Factorials are a vital concept in permutations and combinatorics, especially when dealing with multisets. The factorial of a non-negative integer \( n \), denoted as \( n! \), represents the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are used to calculate the total number of ways to arrange a set of objects.

For counting problems involving repeated elements, factorials help adjust the total count to account for indistinguishable objects. This is essential when determining the number of distinct arrangements in permutations of multisets.

Knowing how to compute factorials, especially for smaller numbers, is an essential step in solving various combinatorial problems.

Symbol Arrangements

Symbol arrangements in permutations involve ordering symbols in distinct sequences. When symbols or objects include repeats, as in a multiset, calculating arrangements requires special consideration.

For instance, if you have symbols such as **#,**, **@**, and **%**, you count the distinct ways to order them by considering the multiple occurrences of some symbols. This is achieved using permutations of multisets.

In a multiset, the formula \( \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \) helps determine distinct arrangements, where \( n \) is the total number of symbols, and \( n_1, n_2, \ldots, n_k \) are the frequencies of each repeating symbol. Understanding how to apply this formula ensures accurate calculation of unique permutations.

Combinatorics

Combinatorics is the branch of mathematics focusing on counting, combination, and permutation of sets. It provides the tools to determine how many ways you can arrange items.

When dealing with permutations, you consider all possible orderings of a set. Combinatorics involves both permutations and combinations: where permutations involve ordered arrangements, and combinations consider selection of items without regard to order.

In problems like finding arrangements of symbols, combinatorics guides you through using factorials and combinations to account for repeated elements. It forms the foundational framework required to tackle various counting problems efficiently.

Counting Principles

Counting principles are fundamental strategies used to solve problems in combinatorics and beyond. They encompass several key ideas, such as the multiplication principle, which explains how to calculate the total number of outcomes in multi-step scenarios.

A primary component of counting principles is the inclusion-exclusion principle, helping to account for overlaps in sets. These principles become particularly useful in complex problems like determining symbol arrangements in multisets.

By using these counting techniques, you ensure that all possible outcomes are considered, adjusting for repetition, and avoiding over-counting. Implementing counting principles provides clarity and precision in tackling various mathematical challenges, ensuring effective problem-solving solutions.