Problem 33 - College Algebra with Corequisite Support

Question

,#,#,@,@,\$,\$,\$,\%,\%,\%,\% # For the following exercises, find the distinct number of arrangements. The symbols in the string #,#,#,@,@,\$,\$,\$,\%,\%,\%,\%

Step-by-Step Solution

Answer

277,200

Step 1: Determine the Total Number of Symbols

First, count the total number of symbols in the arrangement. We have 3 '#', 2 '@', 3 '$', and 4 '%' symbols. Adding these gives the total: \[3 + 2 + 3 + 4 = 12\] symbols in total.

Step 2: Use the Formula for Arrangements of Multisets

To find the number of distinct arrangements of these symbols, we use the formula for permutations of a multiset: \[\frac{n!}{n_1! \times n_2! \times ... \times n_k!}\] where \(n\) is the total number of items and \(n_1, n_2, ..., n_k\) are the frequencies of each distinct item.

Step 3: Compute the Factorials of the Totals

First, calculate the factorial of the total number of symbols: \[12! = 479,001,600\].Now, calculate the factorials of the repeated symbols:\[3! = 6,\] \[2! = 2,\] \[3! = 6,\] and \[4! = 24\].

Step 4: Substitute the Values into the Formula

Substitute these values into the multiset permutation formula:\[\frac{12!}{3! \times 2! \times 3! \times 4!} = \frac{479,001,600}{6 \times 2 \times 6 \times 24}\].

Step 5: Simplify the Expression

Calculate the denominator: \[6 \times 2 \times 6 \times 24 = 1,728\].Now divide the factorial of the total symbols by this product:\[\frac{479,001,600}{1,728} = 277,200\].

Step 6: Verify the Calculation

To double-check, re-evaluate each calculation step: - Ensure the calculations of each factorial are correct. - Check the multiplication and division again to be confident in the final answer.

Key Concepts

Factorials

Factorials are a fundamental part of solving problems involving permutations and combinations. The concept of a factorial is seen notated as an exclamation point (!) after a number, and it represents the product of all positive integers up to that number. For example, 5 factorial, written as 5!, equates to 5 × 4 × 3 × 2 × 1 = 120. In permutations of multisets, factorials help determine the total permutations possible.

Factorials grow very rapidly, which is why they are powerful in calculations involving large sets. In the solution provided, the factorial of 12, denoted as 12!, is calculated by multiplying every number from 1 to 12, resulting in 479,001,600. This is the first step in understanding how many total arrangements can be made from all symbols before considering any repetitions.

Distinct Arrangements

When arranging symbols where some are identical, the goal is to find the distinct arrangements or unique permutations. This is crucial for cases like this exercise, where the symbols are not all different.

In the example: #,#,#,@,@,$,#,#,%,%,%,%, the task is to determine how many unique sequences you can make. We use a formula that divides the factorial of the total number of symbols by the product of the factorials of each group of identical symbols. This accounts for identical items that could otherwise be indistinguishable in their arrangement.

• The multiset permutation formula utilized is: \ [ \frac{n!}{n_1! \ \times \ n_2! \ \times \ ... \ \times \ n_k!} \ ]
• \(n\) is the total number of symbols
• \(n_1,n_2,...,n_k\) are the respective counts of each identical symbol set.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. This field is essential to solve complex problems related to permutations like the one given in the exercise.

One of the main challenges in combinatorics is finding the number of ways items can be selected or ordered—also known as permutations and combinations. When dealing with identical items, as in the provided problem, the entry into permutations of multisets is necessary.
This problem is about arranging multiset elements considering repetitions, an advanced permutation topic that illustrates combinatorial principles in action. By understanding combinatorics, students learn to apply logical steps to solve configuration problems effectively.

Mathematical Symbols

In mathematics, symbols such as factorials, exponents, and brackets are invaluable for clearly expressing complex operations and formulas. For example, the factorial symbol (!) is critical in permutation calculations.

Another key symbol seen in our context is the division and multiplication to denote the operation sequence in the fraction: \[ \frac{12!}{3! \ \times 2! \ \times 3! \ \times 4!}. \] This clearly indicates a division of the total factorial by the product of factorials of each type of symbol, signifying distinctiveness in arrangements.
These mathematical symbols allow us to efficiently communicate complicated ideas in a concise manner. Mastery of these helps students not only in calculations but also in comprehending mathematical instructions. As seen in the solution, breaking down symbols into meaningful operations is part of successfully navigating mathematical results.